exponential and logarithmic functions - the eyethe-eye.eu/public/worldtracker.org/college...

285

CHAPTER 3

Exponential and Logarithmic Functions

ACTIVITY 3.1

The Summer Job

OBJECTIVES

1. Determine the growth ordecay factor of an expo-nential function.

2. Identify the properties ofthe graph of an exponen-tial function defined by

where and.

3. Graph an exponentialfunction.

b � 1b 7 0y � b

x,

CLUSTER 1 Exponential Functions

Your neighbor’s son will be attending college in the fall, majoring in mathemat-ics. On July 1, he comes to your house looking for summer work to help pay forcollege expenses. You are interested since you need some odd jobs done, but youdon’t have a lot of extra money to pay him. He can start right away and will workall day July 1 for 2 cents. This gets your attention, but you wonder if there is acatch. He says that he will work July 2 for 4 cents, July 3 for 8 cents, July 4 for16 cents, and so on for every day of the month of July.

1. Do you hire him?

Yes.

For Problems 2–8, assume that you do hire him.

2. How much will he earn on July 5? July 6?

He will earn $0.32 on July 5 and $0.64 on July 6.

3. What will be his total pay for the first week of July (July 1 through July 7)?

$2.54

4. a. Complete the following table.

1 2

2 4

3 8

4 16

5 32

6 64

7 128

8 256

DAY IN JULY PAY IN CENTS(Input) (Output)

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 285

b. Do you notice a pattern in the output values? Describe how you canobtain the pay on a given day knowing the pay on the previous day.

To obtain the pay on a given day, multiply the previous day’s pay by 2.

c. Use what you discovered in part b to determine the pay on July 9.

The pay on July 9 is cents � $5.12.

5. a. The pay on any given day can be written as a power of 2. Write each payentry in the output column of the table in Problem 4 as a power of 2. Forexample, , .

, , , , , , ,

b. Let n represent the number of days worked. Write an equation for thedaily pay, P1n2, (in cents) as a function of n, the number of days worked.Note that the number of days worked is the same as the July date.

c. Use the equation from part b to determine how much your neighbor’s sonwill earn on July 20. That is, determine the value of P1n2 when .What are the units of measurement of your answer?

cents � $10,485.76

d. How much will he earn on July 31? Be sure to indicate the units of youranswer.

cents � $21,474,836.48

e. Was it a good idea to hire him?

No; I could not afford him.

6. a. Determine the average rate of change of P1n2 as n increases from toWhat are the units of measurement of your answer?

cents/day

b. Determine the average rate of change of P1n2 as n increases from toInclude units in your answer.

cents/day

c. Is the function linear? Explain.

The function is not linear. The average rate of change is not constant.

7. a. What is the practical domain of the function defined by ?

The input, n, is the July date, so the practical domain is wholenumbers to 31.n � 1

P1n 2 � 2n

P18 2 � P17 28 � 7 �

256 � 1288 � 7 �

1281 � 128

n � 8.n � 7

P14 2 � P13 24 � 3 �

16 � 84 � 3 �

81 � 8

n � 4.n � 3

P 131 2 � 2,147,483,648

P 120 2 � 1,048,576

n � 20

P 1n 2 � 2n

256 � 28128 � 2764 � 2632 � 2516 � 248 � 234 � 222 � 21

4 � 222 � 21

2 # 256 � 512

286 Chapter 3 Exponential and Logarithmic Functions

DANFMC03_0321447808 pp3 12/27/06 8:12 PM 86

b. Sketch a scatterplot of ordered pairs of the form 1n, P1n22 from July 1 toJuly 10 on appropriately scaled and labeled axes.

The function defined by gives the relationship between the pay P1n2(in cents) and the given July date, n, worked. This function belongs to a family offunctions called exponential functions.

P 1n 2 � 2n

500

750

1000

250

1 3 5 7 9 11n

Day

Pay

(in

do

llars

)

P(n)

Activity 3.1 The Summer Job 287

Some exponential functions can be defined by equations of the form where the base b is a constant such that b is a positive number not equal to 11 and 2. Such functions are called exponential functions because theindependent variable (input) x is the exponent.

b � 1b 7 0

y � bx,

Example 1 Some examples of exponential functions are

, where b � 10, , where b � 1.08,

, where , and , where b � 0.75.T1x 2 � 10.75 2xb �12

V 1x 2 � a12bx

h1x 2 � 11.08 2xg1x 2 � 10x

Graphs of Exponential Functions

Because n in (the summer job situation) represents a given day in July,the practical domain (whole numbers from 1 to 31) limits the investigation of theexponential function.

8. a. Consider the general function defined by . Use your graphingcalculator to sketch a graph of this function. Use the window Xmin � �10, Xmax � 10, Ymin � �2, and Ymax � 10.

f 1x 2 � 2x

P1n 2 � 2n

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 287

b. Because the graph of the general function is continuous (it hasno holes or breaks), what appears to be the domain of the function f ?What is the range of the function f ?

The domain of f is the set of all real numbers. The range of f is the setof all positive real numbers.

c. Determine the y-intercept of the graph of f by substituting 0 for x in theequation and solving for y.

; therefore, the y-intercept is 10, 12.d. Is the function f increasing or decreasing?

increasing

y � 20 � 1

y � 2x

f 1x 2 � 2x


DEFINITION

If the base b of an exponential function defined by is greater than 1, then bis the growth factor. The graph of is increasing if For each in-crease of 1 of the value of the input, the output increases by a factor of b.

b 7 1.y � bxy � bx

9. Identify the growth factor, if any, for the given function.

a.The growth factor is 1.08.

b.The base is less than 1; therefore, it is not a growth factor.

c.This is not an exponential function; therefore, there is no growthfactor.

d.The growth factor is 10.

10. Return to the graph of

a. Does the graph of appear to have an x-intercept?

No; the graph never touches the x-axis.

b. Use your calculator to complete the following table.

f 1x 2 � 2x

f 1x 2 � 2x.

g1x 2 � 10 x

y � 8x

h1x 2 � 0.8x

y � 1.08x

Example 2 The base 2 of f(x) is the growth factor because each time theinput, x, is increased by 1, the output is multiplied by 2.

� 2x

x �1 �2 �4 �6 �8 �10

f 11x22 � 2x 0.5 0.25 0.0625 0.0156 0.00591 0.000977

Note: is equivalent to 1210 � 0.000977.2�10

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 288

c. As the values of the input variable x decrease, what happens to the outputvalues?

The output values get closer to zero.

d. Use the trace feature of your graphing calculator to trace the graph offor . What appears to be the relationship between the

graph of and the x-axis when x becomes more negative?

The graph of gets closer to the x-axis as x becomes morenegative.

f 1x 2 � 2x

y � 2xx 6 0f 1x 2 � 2x


DEFINITION

A horizontal axis having equation is called a horizontal asymptote of thegraph of a function defined by , where and . The graph of thefunction gets closer and closer to the x-axis as the input gets farther fromthe origin, in the negative direction.

1y � 0 2b � 1b 7 0y � bx

y � 0

Example 3 The x-axis is the horizontal asymptote of and because, as x gets more negative, the graph gets closer and closerto the x-axis. See the graph that follows.

y y = 7x

y = 3x

x2

6

8

2

4

–2–4 4

y � 7xy � 3x


x �3 �2 �1 0 1 2 3 4 5

f 11x22 � 2x 0.125 0.25 0.5 1 2 4 8 16 32

g 11x22 � 10x 0.001 0.01 0.1 1 10 100 1000 10,000 100,000

b. Sketch the graph of the functions f and g on your graphing calculator. Usethe window Xmin � �5, Xmax � 5, Ymin � �2, and Ymax � 9.

Graph at left.

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 289

c. Use the results from parts a and b to describe how the graphs ofand are similar and how they are different. Be sure

to include domain, growth factor, x- and y-intercepts, and horizontalasymptotes. Also discuss whether the graph of g increases faster or slowerthan the graph of f.

The graphs of f and g are both increasing. The domain of each func-tion is the set of all real numbers, and the range is the set of all posi-tive real numbers. The y-intercept is 10, 12 for each graph, and there isno x-intercept. The x-axis is a horizontal asymptote. The graph of gincreases at a faster rate than the graph of f since its growth factor,10, is larger than 2.


g1x 2 � 10 xf 1x 2 � 2x


x �3 �2 �1 0 1 2 3 4 5

8 4 2 1 0.5 0.25 0.125 0.0625 0.03125V (x) � (12) x

b. Describe how you can obtain the output value for , using the outputvalue for .

Multiply V 152 by to get V 162.c. Sketch the graph of . Verify your sketch using your graphing

calculator.

d. What are the domain and range of the function V?

The domain is the set of all real numbers, and the range is the set ofall positive real numbers.

e. Determine the vertical intercept of the graph of V.

10, 12f. Is the function V increasing or decreasing?

decreasing

2 4

–2

–4

4

2

6

8

–2–4–6 6

V(x) =x1–

2

V

x

V 1x 2 � 1122x12

x � 5x � 6

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 290

13. Identify the decay factor, if any, for the given function.

a.The decay factor is 0.98.

b.1.8 is greater than 1, so it is a growth factor.

c.0.8x is a linear function, so there is no growth factor.

d.

The decay factor is

14. Return to the graph of .

a. Does the graph of have an x-intercept?

No.

b. Complete the following table.

V 1x 2 � 1122xV 1x 2 � 1122x27.

g1x 2 � 1272xy � 0.8x

h1x 2 � 1.8x

y � 0.98x


DEFINITION

If the base b of an exponential function is between 0 and 1, then b is thedecay factor. The graph of is decreasing if . For each increaseof 1 of the value of the input, the output decreases by a factor of b.

0 6 b 6 1y � bxy � bx

Example 4 The base in the function is the decay factor because

each time x is increased by 1, the output value is multiplied by 12.

V (x) � (12)x12

x 1 3 5 7 10

0.5 0.125 0.0313 0.00781 0.000977V (x) � (12)x

c. As the values of the input variable x get larger, what happens to the outputvalues?

The output values get closer to zero.

d. Does the graph of V have a horizontal asymptote? Explain.

Yes; the horizontal asymptote is the x-axis 1y � 02.

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 291

b. Without graphing, how might you determine which of the functions inpart a increase and which decrease? Explain.

If the base is greater than 1, the graph is increasing. If the base isbetween 0 and 1, the graph is decreasing.

16. Examine the output pattern to determine which of the following data sets islinear and which is exponential. For the linear set, determine the slope. Forthe exponential set, determine the growth or decay factor.

a.


1.08 growth none 10, 12 y � 0 increasing

0.75 decay none 10, 12 y � 0 decreasing

3.2 growth none 10, 12 y � 0 increasing

decay none 10, 12 y � 0 decreasing14r 1x 2 � 1142x

f 1x 2 � 13.2 2xT 1x 2 � 10.75 2xh1x 2 � 11.08 2x

FUNCTION BASE, b GROWTH OR x-INTERCEPT y-INTERCEPT HORIZONTAL INCREASINGDECAY FACTOR ASYMPTOTE OR DECREASING

x �2 �1 0 1 2 3 4

y �8 �4 0 4 8 12 16

This data is linear; the slope is 4.

b. x �2 �1 0 1 2 3 4

y 1 4 16 64 25614

116

This data is exponential; the growth factor is 4.

17. Determine the decay factor of the function represented by the data, andcomplete the table.

x �2 �1 0 1 2

f 11x22 16 4 1 0.25 0.0625

The decay factor is 14

Functions defined by equations of the form , where and ,are called exponential functions and have the following properties.

1. The domain is all real numbers.

2. The range is .y 7 0

b � 1b 7 0y � bxSUMMARYACTIVITY 3.1

15. a. For each of the following exponential functions, identify the base, b, anddetermine whether the base is a growth or decay factor. Graph eachfunction on your graphing calculator, and complete the table below.

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 292

Exercise numbers appearing in color are answered in the Selected Answers appendix.


3. If , the function is decreasing and has the following general shape.

In this case, b is called the decay factor.

4. If , the function is increasing and has the following general shape.

In this case, b is called the growth factor.

5. The vertical intercept (y-intercept) is 10, 12.6. The graph does not intersect the horizontal axis. There is no x-intercept.

7. The line (the x-axis) is a horizontal asymptote.y � 0

b 7 1

0 6 b 6 1

EXERCISESACTIVITY 3.1

1. a. Complete the following tables.

x �3 �2 �1 0 1 2 3

0.008 0.04 0.2 1 5 25 125h(x) � 5x

x �3 �2 �1 0 1 2 3

125 25 5 1 0.2 0.04 0.008g(x) � (15) x

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 293

c. Use the tables and graphs in parts a and b to complete the following table.


5 growth none 10, 12 y � 0 increasing

decay none 10, 12 y � 0 decreasing15g1x 2 � 1152 x

h1x 2 � 5x

FUNCTION BASE, b GROWTH OR x-INTERCEPT y-INTERCEPT HORIZONTAL INCREASING ORDECAY FACTOR ASYMPTOTE DECREASING


x �3 �2 �1 0 1 2 3

0.037 0.111 0.333 1 3 9 27

�27 �8 �1 0 1 8 27

�9 �6 �3 0 3 6 9h(x) � 3x

g(x) � x 3

f (x) � 3 x

b. Sketch a graph of each of the given functions f, g, and h.

c. Describe any similarities or differences that you observe in the graphs.

All the graphs are increasing. All have the domain of all real numbers.increases fastest. has a horizontal asymptote

.1y � 0 2 f 1x 2 � 3xf 1x 2 � 3x

f(x) = 3x

1

–20

–10

–30

20

10

30

–1–2–3 2 3

h(x) = 3x

g(x) = x 3

x

y

b. Sketch graphs of h and g on the following grid.

–1

g(x) = x1–

5 h(x) = 5x

1 2

48

1216202428

–2

y

x

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 294

This data is exponential with a growth factor of 3.

b.

3. Using your graphing calculator, investigate the graphs of the following fami-lies (groups) of functions. Describe any relationships within each family, in-cluding domain and range, growth or decay factors, vertical and horizontalintercepts, and asymptotes. Identify the functions as increasing or decreasing.

a. ,

The graph of f is a decreasing exponential function with a decay fac-tor of The graph of g is an increasing exponential function with agrowth factor of The graphs of f and g are reflections in the y-axis.

b. ,

The graph of f is an increasing exponential function with a growthfactor of 10. The graph of g is the graph of f reflected in the x-axis.

c. ,

The graph of f is an increasing exponential function with a growthfactor of 3. The graph of g is a decreasing exponential function witha decay factor of . The graphs of f and g are reflections in the y-axis.

4. Determine which of the following data sets are linear and which are expo-nential. For the linear sets, determine the slope. For the exponential sets,determine the growth factor or the decay factor.

a.

13

g1x 2 � 1132xf 1x 2 � 3x

g1x 2 � �10xf 1x 2 � 10 x

43.

34.

g1x 2 � a43bx

f 1x 2 � a34bx


x �2 �1 0 1 2 3 4

y 1 3 9 27 8113

19

x �2 �1 0 1 2 3 4

y 2 2.5 3 3.5 4 4.5 5

x �2 �1 0 1 2 3 4

y 0.75 1.5 3 6 12 24 48

x �2 �1 0 1 2 3 4

y 6.25 2.5 1 0.4 0.16 .064 .0256

This data is exponential with a growth factor of 2.

d.

This data is linear with a slope of 0.5.

c.

This data is exponential with a decay factor of 0.4.

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 295

b. Would you expect to decrease faster or slower thanfor ? Explain.

The function f will decrease faster than g because its decay factor issmaller.

7. Determine the domain and range of each of the following functions.

a.

The domain is all real numbers; the range is y 7 0.

2

4

6

8

10

4 6 82

y

x

x 7 0g1x 2 � 10.70 2x f 1x 2 � 1122x


5. Assume that y is an exponential function of x.

a. If the growth factor is 1.08, then complete the following table.

x 0 1 2 3

y 23.1 24.948 26.944 29.099

b. If the decay factor is 0.75, then complete the following table.

x 0 1 2 3

y 10 7.5 5.625 4.21875

6. a. Would you expect to increase faster or slower than for Explain. (Hint: You may want to use your graphing calculatorfor help.)

The function f will increase faster than g because its growth factor islarger.

x 7 0?g1x 2 � 2.5xf 1x 2 � 3x

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 296

b.

The domain is all real numbers; the range is .

8. Take a piece of paper from your notebook. Let x represent the number oftimes you fold the paper in half and represent the number of sectionsthe paper is divided into after the folding.

a. Complete the table of values.

f 1x 2

y 7 0

2

4

6

8

10

4 6 82

y

x


x 0 1 2 3 4 5

f 11x22 1 2 4 8 16 32

b. If you could fold the paper 8 times, how many individual sections willthere be on the paper?

256

c. Does this data represent an exponential function? Explain.

The data is exponential with a growth factor of 2.

d. What is the practical domain and range in this situation?

(Answers may vary.) The practical domain is the set of nonnegative in-tegers from 0 to 10. The practical range is the set of whole-numberpowers of 2 up to 210.

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 297


ACTIVITY 3.2

Cellular Phones

OBJECTIVES

1. Determine the growthand decay factor for anexponential function rep-resented by a table of val-ues or an equation.

2. Graph exponential func-tions defined by ,where and

3. Determine the doublingand halving time.

a � 0.b � 1,b 7 0

y � abx

During a meeting, you hear the familiar ring of a cell phone. Without hesitation,several of your colleagues reach into their jacket pockets, briefcases, and pursesto receive the anticipated call. Although sometimes annoying, cell phones havebecome part of our way of life.

The following table shows the rapid increase in the number of cellular phones(figures are approximate) in the late 1990’s. Note that the input variable (year) in-creases in steps of 1 unit (year).

1996 44.248

1997 55.312

1998 69.140

1999 86.425

2000 108.031

YEAR NUMBER OF CELLULAR PHONESAS OF JAN. 1 (in millions)

Calling All Cells

1. Is this a linear function? How do you know?

This is not a linear function; the rate of change is not constant.

2. a. Evaluate the indicated ratios to complete the following table.

1.25 1.25 1.25 1.25

No. of phones in 1997 No. of phones in 1998 No. of phones in 1999 No. of phones in 2000No. of phones in 1996 No. of phones in 1997 No. of phones in 1998 No. of phones in 1999

b. What do you notice about the values of the table?

All of the values are the same, 1.25.

In an exponential function with base b, equally spaced input values yield outputvalues whose successive ratios are constant. If the input values increase by incre-ments of 1, the common ratio is the base b. If , b is the growth factor; if

, b is the decay factor.0 6 b 6 1b 7 1

3. a. Does the relationship in the table preceding Problem 1 represent anexponential function? Explain.

Yes, the successive ratios are constant.

b. What is the growth factor?

The growth factor is 1.25.

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 298

c. As a consequence of the result found in part b, you can start with 44.248,the number of cellular phones in 1996, and obtain the number of cellularphones in 1997 by multiplying by the growth factor, b � 1.25. You canthen determine the number of cellular phones in 1998 by multiplying thenumber of cellular phones in 1997 by b, and so on. Verify this with yourcalculator. Note that because the exponential function is a mathematicalmodel, the results will vary slightly from the actual number of phonesgiven in the table preceding Problem 1.

Once you know the growth factor you can determine the equation thatgives the number of phones as a function of t, the number of years since 1996.Note that corresponds to 1996, to 1997, and so on.

4. a. Complete the table.

t � 1t � 0

1b � 1.25 2,

Activity 3.2 Cellular Phones 299

0 44.248 44.24811.2520 44.248

1 144.24821.25 44.24811.2521 55.31

2 44.24811.2522 69.1375

3 44.24811.2523 86.421875144.248 # 1.25 # 1.25 21.25

144.248 # 1.25 21.25

t CALCULATION FOR THE EXPONENTIAL FORM NUMBER OF CELL PHONESNUMBER OF CELL PHONES

b. Use the pattern in the preceding table to help you write the equation of theform , where N1t2 represents the number of cell phones (inmillions) in use at time t, the number of years since 1996.

c. What is the practical domain of the function N?

0 through 4

d. Graph the function N on your graphing calculator, and then sketch theresult below on an appropriately scaled and labeled axis.

N(t) = 44.248(1.25)t

75

100

25

50

10 2 3 4 5t

Number of Years since 1996

Nu

mb

er o

f C

ell P

ho

nes

(in

mill

ion

s)

N

N1t 2 � 44.24811.25 2tN1t 2 � a # bt

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 299

e. Determine the vertical intercept of the graph of N by substituting 0 for theinput, t. What is the practical meaning of the vertical intercept in thissituation?

The vertical intercept is 10, 44.2482. There were 44.248 million cellphones in 1996 1t � 0 2.


DEFINITION

Many exponential functions can be represented symbolically by where a is the value of f when and b is the growth or decay factor. If the in-put, t, of represents time, then the coefficient a is called the initial value.y � a # bt

t � 0f 1 t 2 � a # bt,

Example 1 The exponential function defined by has y-intercept (0, 5) and growth factor . The exponential function definedby has y-intercept and decay factor

5. Use the function defined by to estimate the number ofcell phone users in 2005. Do you think this is a good estimate? Explain.

. The function predicts 329 million cellphone users in 2005 This is not reliable because it is well outsideof the original data. According to the U.S. Census Bureau, the actualnumber of cell phone users in 2005 was 196 million.

6. a. Use the graph of the exponential function and thetrace or table feature of your graphing calculator to estimate the numberof years it takes for the number of cell phone users to double from44.248 million to 88.496 million.

It takes about 3.1 years for the number of cell phone users to double.

b. Estimate the time necessary for the number of cell phone users to doublefrom 88.496 million to 176.992 million. Verify your estimate using yourcalculator.

about 3.1 yr.

c. How long will it take for any given number of cell phone users to double?

about 3.1 yr.

N1t 2 � 44.24811.25 2t

1t � 9 2.N19 2 � 44.24811.25 29 � 329.67

N1t 2 � 44.24811.25 2tb � 0.75.

(0, 12)h(x) � 12 (0.75)x

b � 2f(x) � 5 # 2x

DEFINITION

The doubling time of an exponential function is the time it takes for an output todouble. The doubling time is determined by the growth factor and remains thesame for all output values.

Example 2 The balance B (t), in dollars, of an investment account is defined by, where t is the number of years. The initial

value for this function is $5500. Determine the value of t when thebalance is doubled or equal to $11,000.

B( t) � 5500(1.12)t

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 300

SOLUTION

If you use the table feature of your calculator, the doubling time is estimated at 6.1years (see the following calculator graphic). The intersect feature on the graphingcalculator shows the doubling time to be 6.12 years to the nearest hundredth.

Decreasing Exponential Functions, Decay Factor,and Halving Time

You have just purchased a new automobile for $22,000. Much to your dismay, youhave just learned that you should expect the value of your car to depreciate by30% per year! The following table shows the book value of the car for the nextseveral years, where V is the value in thousands of dollars.

When a quantity is increased or decreased by a constant percent rate, it can bemodeled by an exponential function. In this situation, the car value is decreasedby 30% per year. A decreasing exponential function has a decay factor, b, with

. For consecutive values of the input, an output value is determined bymultiplying the previous output value by b.

7. As the input, t, increases from 0 to 1, the output, V, decreases from 22 to 15.4(in $1000).

a. Determine the value that 22 is multiplied by to get 15.4.

15.4 � 22 � 0.7

b. Use the result from part a to complete the following table.

0 6 b 6 1

022

115.4

210.8

37.5

45.3

t (year)

V(t)(in thousands

of dollars)

Depreciation: Taking Its Toll


0 22 2210.720 22

1 2210.72 2210.721 15.4

2 2210.72 10.72 2210.722 10.8

3 2210.72 10.72 10.72 7.5

4 2210.72 10.72 10.72 10.72 5.32210.7 242210.7 23

tt CALCULATION OF THE VALUE OF EXPONENTIAL VALUE (in $1000)THE CAR FORM

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 301

c. Use the pattern in the preceding table to write an equation in the formthat gives the car value, V, as a function years, t.

d. Input the function into on your calculator and sketch the result below.

e. Determine the value of the vertical intercept of the graph. Write the resultas an ordered pair.

10, 222f. What is the practical meaning of the vertical intercept in this situation?

The car cost $22,000 when it was new 1t � 02.8. a. Estimate the number of years, t, it takes for the value of the car to be

$11,000, half the original value. (Hint: Put 11000 in and find the pointof intersection.)

t � 1.94 yr.

b. How many years will it take for the car value to be halved again, that isfrom $11,000 to $5500?

another 1.94 yr.

Y2

Y1

V � 2210.7 2tV � a # bt


DEFINITION

The half-life of an exponential function is the time it takes for an output to decayby one-half. The half-life is determined by the decay factor and remains the samefor all output values.

Example 3 The population of Buffalo, New York, can be modeled by the equa-tion B(t) � 1102(0.995)t with t � 0 representing the year 1970 andB(t) representing the population in thousands. If the populationof Buffalo continues to decline at the same rate, determine thenumber of years it will take for the population of Buffalo to be onehalf of the 1970 population.

SOLUTION

The equation indicates that the population of Buffalo, NY, in 1970 1t � 02 is B102 � 110210.99520 � 1102 thousand people. Therefore, half of that is 551 thou-sand people. Use the table feature of your graphing calculator; the halving time isestimated at 138 years.

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 302

Use the intersect feature of your graphing calculator and you will also find thehalving time to be 138 years rounded to the nearest year.


1. For exponential functions defined by , a is the value of f when(sometimes called the initial value), and b is the growth or decay factor.

2. The vertical intercept of these functions is 10, a2.

3. In an exponential function, equally spaced input values yield output valueswhose successive ratios are constant. If the input values increase by 1 unit,then

a. the constant ratio is the growth factor if the output values are increasing

b. the constant ratio is the decay factor if the output values are decreasing

4. The doubling time of an increasing exponential function is the time it takesfor an output to double. The doubling time is set by the growth factor andremains the same for all output values.

5. The half-life of a decreasing exponential function is the time it takes for anoutput to decay by one-half. The half-life is determined by the decay factorand remains the same for all output values.

x � 0f 1x 2 � abxSUMMARY

ACTIVITY 3.2


1. The population of Russia in selected years can be approximated by the followingtable.


YEAR 1995 1996 1997 2000

POPULATION (in millions) 148.0 147.6 146.9 146.0

a. Let 1995 correspond to . Let b be the ratio between the population ofRussia in 1996 and 1995. Determine an exponential function of the form

to represent the population of Russia symbolically. Round to fourdecimal places.

b. Does the function in part a give an accurate value of the population of Russiain 2000? Explain.

Yes; substituting 5 for t yields .

c. Use your model in part b to predict the population of Russia in 2007.

millionP � 148.010.9973 212 � 143.3

P � 148.010.9973 25 � 146.01

P � 148.010.9973 2t

y � a # bt

t � 0

DANFMC03_0321447808 pp3 1/6/07 1:20 PM Page 303

2. Without using your graphing calculator, match each graph with its equation.Then check your answer using your graphing calculator.

a. b.

i. ii.

a. ii b. i

3. Which of the following tables represent exponential functions? Indicate thegrowth or decay factor for the data that is exponential.

a.

x

y

x

y

g1x 2 � 311.73 2xf 1x 2 � 0.510.73 2x


x 0 1 2 3 4

y 0 2 16 54 128

This data is not exponential. The successive ratios are not constant.

b. x 0 1 2 3 4

y 1 4 16 64 256

This data is exponential. The growth factor is .

c.

b � 4

x 1 2 3 4 5

y 1750 858 420 206 101

This data is exponential. The decay factor is .

4. a. Sketch a graph of and on the same coordinateaxes.

–2

–2

–4

f(x) = 2x

g(x) = 3 • 2x

2 4

2

4

6

y

x

g1x 2 � 3 # 2xf 1x 2 � 2x

b � 0.49

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 304

b. Describe how the graphs of f and g are similar and how they are different.

For the same value of x, the graph of g is 3 times the distance fromthe x-axis that the graph of f is.

5. If , determine the exact value of each of the following, whenpossible. Otherwise, use your calculator to approximate the value to thenearest hundredth.

f 1x 2 � 3 # 4x


c. d.

f 11.3 2 � 3 # 41.3 � 18.1886

f 11.3 2f 12 2 � 3 # 42 � 3 # 16 � 48

f 12 2

a.

f 1�2 2 � 3 # 4�2 � 3 # 142 �

316

f 1�2 2

6. In 1995, the United States emitted approximately 1400 million tons ofcarbon into the atmosphere. This represented about one-fourth of the worldtotal. The U.S. emissions were increasing at about 1.3% per year. If t repre-sents the number of years since 1995 and represents the amount of car-bon (in millions of tons) emitted in a given year, then .

a. Complete the following table.

A1t 2 � 140011.013 2tA1t 2

t, NUMBER OF 0 1 2 3 4 5YEARS SINCE 1995

A 11t22 , AMOUNT OF U.S. CARBON 1400 1418 1437 1455 1474 1493EMISSIONS (in millions of tons)

b. Determine the growth factor for carbon emissions.

The growth factor is .

c. Sketch a graph of this exponential function. Use and.

d. Use the equation to determine the amount of carbonemission in 2010. Include the units of measurement in your answer.

In 2010, , so million tons ofcarbon.

A115 2 � 1400 # 1.01315 � 1699t � 15

A1t 2 � 140011.013 2t

A(t) = 1400(1.013)t

500

1000

1500

2000

2500

5 10 15 20 25t

A

0 � A1t 2 � 25000 � t � 25

b � 1.013

b.

f 1122 � 3 # 41>2 � 3 # 2 � 6

f 1122

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 305

e. Use the graph and trace features of your graphing calculator to approximatethe year in which carbon emissions in the United States will exceed 2000million tons.

The carbon emissions will exceed 2000 million tons in 2022.

7. Chlorine is used to disinfect swimming pools. The chlorine concentrationshould be between 1.5 and 2.5 parts per million (ppm). On sunny, hot days,30% of the chlorine dissipates into the air or combines with other chemicals.Therefore, chlorine concentration, (in parts per million) in a pool afterx sunny days can be modeled by

a. What is the initial concentration of chlorine in the pool?

2.5 ppm

b. Complete the following table.

A1x 2 � 2.510.7 2x.A1x 2,

11995 � 27 2


x 0 1 2 3 4 5

A 11x22 2.5 1.75 1.225 0.8575 0.6003 0.4202

c. Sketch the graph of the chlorine function.

d. What is the chlorine concentration in the pool after 3 days?

ppm

e. Approximate graphically and numerically the number of days beforechlorine should be added.

Chlorine should be added in 1.4 days.

8. In the nineties, there was a rapid growth in the number of investment clubsin the United States. An investment club is a group of people who meet on aregular basis to invest in the stock market. By joining a club, members areable to share in a diverse portfolio and therefore reduce the risk of losingmoney.

A13 2 � 2.510.70 23 � 0.8575

3

A(x) = 2.5(0.7)x

1

2

20 4 6 8x

Days

Ch

lori

ne

Co

nce

ntr

atio

n(p

pm

)

A

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 306


YEAR NUMBER OF CLUBS AS OF JAN. 1

1990 5820

1991 7180

1992 8860

1993 10,930

1994 13,480

1995 16,630

1996 20,510

a. Does the relationship in the table represent an exponential function?Explain.

The data is exponential; the consecutive ratios of outputs are constant.


The growth factor is

c. Determine the equation that gives the number of clubs, N 1t2, as a function oft, the number of years since 1990. Note that corresponds to 1990.

d. Graph the function.

e. What is the vertical intercept? What is the practical meaning of thisintercept in this situation?

The vertical intercept is . The number ofclubs in 1990 is 5820.

N1t 2 � 582011.23 20 � 5820

6000

12,000

18,000

24,000

3000

9000

15,000

21,000

N(t) = 5280 • (1.23)t

21 3 4 5 6 7t

Years since 1990

Nu

mb

er o

fIn

vest

men

t C

lub

s

y = N(t)

N1t 2 � 582011.23 2tt � 0

b � 1.23.

The following table shows the rapid growth in the number of clubs from1990 to 1996 (figures for the number of clubs are approximate). Note thatthe input variable (year) increases in steps of 1 unit.

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 307

f. Use the equation to estimate the number of clubs in 2000. Do you thinkthis is a good estimate? Explain.

In 2000, , so . This is probably nota good estimate. The growth rate of 23% per year will probably notcontinue.

g. Use the graph of the exponential function and the trace or table feature ofyour grapher to estimate the number of years it takes for the number ofclubs to double from 5820 to 11,640.

According to the model, the number of clubs will double in 3.35 years.

h. How long will it take for any given number of clubs to double?

3.35 yr.

9. Homemade chocolate chip cookies lose their freshness over time. Let the tastequality be 1 when the cookies are fresh. The taste quality decreases accordingto the function

,

where x is the number of days since the cookies were baked.

Determine when the taste quality will be one-half of its value. Use the inter-sect feature of your calculator to determine when f 1x2 is of 1 or 0.5.

The half-life of the freshness of the chocolate chip cookies is 3.1 days.

12

Q � 0.8x

N110 2 � 582011.23 210 � 46,129t � 10


DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 308

According to the 2000 U.S. Census, the city of Charlotte, North Carolina, had apopulation of approximately 541,000.

1. a. Assuming that the population increases at a constant rate of 3.2%, deter-mine the population of Charlotte (in thousands) in 2001.

In 2001, the population will be.

b. Determine the population of Charlotte (in thousands) in 2002.

In 2002, the population will be.

c. Divide the population in 2001 by the population in 2000, and record thisratio.

.

d. Divide the population in 2002 by the population in 2001, and record thisratio.

e. What do you notice about the ratios in parts c and d? What do these ratiosrepresent?

The ratios are equal; they represent the growth factor, 1.032.

576,178 � 558,312 � 1.032.

558,312 � 541,000 � 1.032

558,312 � 0.032 # 558,312 � 558,312 � 17,866 � 576,178

541,000 � 0.032 # 541,000 � 541,000 � 17,312 � 558,312

Activity 3.3 Population Growth 309

ACTIVITY 3.3

Population Growth

OBJECTIVES

1. Determine annual growthor decay factor of an ex-ponential function repre-sented by a table ofvalues or an equation.

2. Graph an exponentialfunction having equation

, a � 0.y � a11 � r 2 x

Linear functions represent quantities that change at a constant rate (slope). Exponen-tial functions represent quantities that change by a constant growth or decay factor.

Example 1 Population growth, sales and advertising trends, compound inter-est, spread of disease, and concentration of a drug in the blood areexamples of quantities that increase or decrease by a constantgrowth factor.

2. a. Let t represent the number of years since 2000 (t � 0 corresponds to 2000).Use the growth factor from Problem 1 to complete the following table.

Charlotte

t, YEARS(since 2000)

P, POPULATION(in thousands)

0 1 2 3

541 558.3 576.2 594.6

4 5

613.6 633.3

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 309

Once you know the growth factor, b, and the initial value, a, you can writethe exponential equation. In this situation, the initial value is the popula-tion, in 1000s, in 2000 1t � 02 and the growth factor is b � 1.032.

b. Write the exponential equation, , for the population of Charlotte.

The growth factor is b � 1.032 and the growth rate is r � 3.2%.

3. a. Write the growth rate, r � 3.2%, as a decimal.

r � 0.032

b. Add 1 to the decimal form of the growth rate r.

1 � r � 1 � 0.032 � 1.032

P � 541 # 1.032t

P � a # bt


t CALCULATION FOR POPULATION EXPONENTIAL FORM P 11t22, POPULATION(in thousands) (in thousands)

0 541 541

1 154121.032 558.3

2 15412 11.0322 11.0322 576.2

3 15412 11.0322 11.0322 11.0322 594.654111.032 2354111.032 2254111.032 2154111.032 20

The growth factor, b, is determined from the growth rate, r, by writing r in decimalform and adding 1: b � 1 � r.

Example 2 Determine the growth factor, b, for a growth rate of r � 8%.

SOLUTION

r � 8% � 0.08, b � 1 � r � 1 � 0.08 � 1.08

c. Solve the equation for the growth factor, b � 1 � r, for r.

r � b � 1

The growth rate, r, is determined from the growth factor, b, by subtracting 1 fromb and writing the result in percent form.

Example 3 Determine the growth rate, r, for a growth factor of b � 1.054.

SOLUTION

r � b � 1 � 1.054 � 1 � 0.054 � 5.4%

The growth rate is 5.4%


DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 310

Example 4

a. Determine the growth factor and the growth rate of the function defined by

SOLUTION

The growth factor is the base 1.7. To determine the growth rate, solve theequation for r.

, or 70%

b. If the growth rate of a function is 5%, determine the growth factor.

SOLUTION

If % or 0.05, the growth factor is .

5. a. Determine the growth factor in the Charlotte population function.

The growth factor is 1.032.

b. Determine the growth rate. Express your answer as a percent.

The growth rate is 0.032 � 3.2%.

6. a. Using the function defined by , determine thepopulation of Charlotte in 2006. That is, determine when .

In 2006, thousand people.

b. Graph the population function with your graphing calculator. Set the win-dow to Xmin � �50, Xmax � 100, Ymin � 0, and Ymax � 13,000, anddisplay the graph.

c. Determine P102. What is the graphical and the practical meaning of P102?. 10, 5412 is the vertical intercept. 541,000 is

the population of Charlotte in 2000 (when ).t � 0P 10 2 � 54111.032 20 � 541

P16 2 � 54111.032 26 � 653.5

t � 6P1t 2P1t 2 � 54111.032 2t

P1t 2 � 54111.032 2t

1 � r � 1 � 0.05 � 1.05r � 5

r � 0.7

1 � r � 1.71 � r

f (x) � 250 (1.7)x.


The equation has the general form where ris the annual growth rate, is the growth factor or the base, b, of theexponential function, t is the time in years, and is the initial value, the popula-tion when t � 0.

P0

11 � r 2P � P011 � r 2 t,P 1 t 2 � 541 11.032 2 t

b. Use the pattern in the table in part a to help you write the equation forP1t2, the population of Charlotte (in thousands), using t, the number ofyears since 2000, as the input value. How does your result compare to theequation obtained in Problem 2?

; the equations are the sameP 1t 2 � 54111.032 2t

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 311

7. a. Use the model to predict Charlotte’s population in 2010.

thousand

b. Verify your prediction on the graph.

8. a. Use the graph to estimate when Charlotte’s population will reach 700,000,assuming it continues to grow at the same rate. Remember, P1t2 is thenumber of thousands.

The population will be 700,000 in approximately 8.18 years, or in theyear 2008.

b. Evaluate P1322 and describe what it means.

If it continues to grow at the same rate, the populationof Charlotte will be 1,482,000 in 2032

9. Use the population model to estimate the population of Charlotte in 2002 andin 2020. In which prediction are you more confident? Why?

2002:

2020: thousand

I am more confident in the estimate for 2002. The growth rate willprobably not remain constant for 20 years.

10. a. Assuming the growth rate remains constant, how long will it take for thepopulation of Charlotte to double its 2000 population?

when

b. Explain how you reached your conclusion in part a.

I used the table feature of the calculator.

Wastewater Treatment Facility

You are working at a wastewater treatment facility. You are presently treatingwater contaminated with 18 micrograms (�g) of pollutant per liter. Your processis designed to remove 20% of the pollutant during each treatment. Your goal is toreduce the pollutant to less than 3 micrograms per liter.

11. a. What percent of pollutant present at the start of a treatment remains at theend of the treatment?

Twenty percent of the pollutant is removed, so 80% remains.

b. The concentration of pollutant is 18 micrograms per liter at the start of thefirst treatment. Use the result of part a to determine the concentration ofpollutant at the end of the first treatment.

The concentration at the end of the first treatment ismg/l.

c. Complete the following table. Round the results to the nearest tenth.

18 # 0.80 � 14.4

t � 22P � 54111.032 2t � 1082

P 120 2 � 54111.032 220 � 1015.76

P 12 2 � 54111.032 22 � 576,000

1t � 32 2.P 132 2 � 1482.

P 110 2 � 54111.032 210 � 741.3


n, NUMBER OF TREATMENTS 0 1 2 3 4 5

C 11n22 , CONCENTRATION OF POLLUTANT, 18 14.4 11.5 9.2 7.4 5.9IN Mg/l, AT THE END OF THE nTH TREATMENT

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 312

d. Write an equation for the concentration, C 1n2, of the pollutant as afunction of the number of treatments, n.

C1n 2 � 1810.80 2n


c. What is the vertical intercept? What is the practical meaning of the inter-cept in this situation?

The vertical intercept is The initial concentra-tion is 18 micrograms per liter.

d. Reset the window of your graphing calculator to Xmin � �5,Xmax � 15, Ymin � �10, and Ymax � 50. Does the graph have ahorizontal asymptote? Explain what this means in this situation.

The horizontal asymptote is The concentration of contami-nates approaches 0 as the number of treatments increases.

C � 0.

C10 2 � 1810.80 20 � 18.

The equation has the general form where r isthe decay rate, is the decay factor or the base of the exponential func-tion, n is the number of treatments, and is the initial value, the concentrationwhen n � 0.

C0

11 � r 2C � C011 � r 2n,C 1n 2 � 18 10.80 2n

Example 5 a. Determine the decay factor and the decay rate of the functiondefined by

SOLUTION

The decay factor is the base, 0.43. To determine the decay rate, solve theequation for r.

, or 57%

b. If the decay rate of a function is 5%, determine the decay factor.

SOLUTION

If , or 0.05, the decay factor is

12. a. If the decay rate is 2.5%, what is the decay factor?

The decay factor is

b. If the decay factor is 0.76, what is the decay rate?

The decay rate is , or 24%.

13. a. Use the function defined by to predict the concentrationof contaminants at the wastewater treatment facility after seven treatments.

After seven treatments, the concentration of the contaminants willbe micrograms per liter.

b. Sketch a graph of the concentration function on your graphing calculator.Use the table in Problem 11c to set a window. Does the graph look likeyou expected it would? Explain.

The graph, at left, looks like a decreasing exponential function.

C17 2 � 18 # 10.80 27 � 3.77

C1n 2 � 1810.8 2n1 � 0.76 � 0.24

b � 1 � 0.025 � 0.975.

1 � r � 1 � 0.05 � 0.95.r � 5%

r � 0.57

1 � r � 0.431 � r

h(x) � 123(0.43)x.

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 313


1. Exponential functions are used to describe phenomena that grow or decayby a constant percent rate per unit time.

2. If r represents the annual growth rate, the exponential function thatmodels the quantity, P, can be written as

where is the initial amount, t represents the amount of elapsed time, andis the growth factor.

3. If r represents the annual decay rate, the exponential function that modelsthe amount remaining can be written as

where is the decay factor.1 � r

P1t 2 � P011 � r 2t,

1 � rP0

P1t 2 � P011 � r 2t,

SUMMARYACTIVITY 3.3



1. Determine the growth and decay factors and growth and decay rates in thefollowing tables.

GROWTH FACTOR GROWTH RATE

1.02 2%

1.029 2.9%

2.23 123%

1.34 34%

1.0002 .02%

DECAY FACTOR DECAY RATE

0.77 23%

.32 68%

0.953 4.7%

.803 19.7%

0.9948 .52%

2. The 2000 U.S. Census reports the populations of Bozeman, Montana, as 27,509and Butte, Montana, as 32,370. Since the 1990 Census, Bozeman’s populationhas been increasing at approximately 1.96% per year. Butte’s population hasbeen decreasing at approximately 0.29% per year. Assume that the growth anddecay rates stay constant.

a. Let P 1t) represent the population t years after 2000. Determine the exponen-tial functions that model the populations of both cities.

Bozeman:

Butte:

b. Use your models to predict the populations of both cities in 2005.

Bozeman:

Butte: P 15 2 � 32,37010.9971 25 � 31,903

P 15 2 � 27,50911.0196 25 � 30,313

P 1t 2 � 32,37010.9971 2tP 1t 2 � 27,50911.0196 2t

14. Use the table or trace feature of your graphing calculator to estimate the num-ber of treatments necessary to bring the concentration of pollutant below 3micrograms per liter.

The concentration will be below 3 micrograms per liter after the ninthtreatment.

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 314

c. Estimate the number of years necessary for the population of Bozeman,Montana, to double.

The population of Bozeman will be 55,018 when

d. Using the table and/or graphs of these functions, predict when the popula-tions will be equal.

The populations will be equal when

3. You have just taken over as the city manager of a small city. The personnelexpenses were $8,500,000 in 2007. Over the previous 5 years, the personnelexpenses have increased at a rate of 3.2% annually.

a. Assuming that this rate continues, write an equation describing personnelcosts, in millions of dollars, where corresponds to 2007.

personnel costs million

b. Sketch a graph of this function up to the year 2017

c. What are your projected personnel costs in the year 2012?

The projected costs in 2012 are million.

d. What is the vertical intercept? What is the practical meaning of the inter-cept in this situation?

The vertical intercept is 10, 8.52. The 2007 personnel costs were8.5 million dollars.

e. In what year will the personnel expenses be double the 2007 personnelexpenses?

The personnel costs will double in approximately 22 years, in 2029.

4. According to the U.S. Bureau of the Census, the population of the UnitedStates from 1930 to 2000 can be modeled by wheret represents the number of years since 1930.

P1t 2 � 120.6 # 1.0125t,

1t � 0 2

C15 2 � 8.511.032 25 � $9.95

//

C(t) = 8.5(1.032)t

8

11

14

17

5 10 15t

Years since 2007

Per

son

nel

Co

sts

(in

mill

ion

s o

f d

olla

rs)

C(t)

1t � 10 2.

C1t 2 � 8.511.032 2tt � 0C1t 2,

t � 7.3.

t � 35.7.


DANFMC03_0321447808 pp3 12/27/06 8:12 PM 5

a. Sketch a graph of the U.S. population model from 1930 to 2000.

b. Determine the annual growth rate and the growth factor from the equation.

The growth factor is b � 1.0125; the growth rate is 0.0125 � 1.25%.

c. Use the population equation to determine the population (in millions) ofthe United States in 2000. How does your answer compare to the actualpopulation of 281.4 million?

million. The prediction is a littlehigher.

5. You have recently purchased a new car for $20,000 by arranging financingfor the next 5 years. You are curious to know what your new car will beworth when the loan is completely paid off.

a. Assuming that the value depreciates at a constant rate of 15%, write anequation that represents the value, V 1t2, of the car t years from now.

b. What is the decay rate in this situation?

The decay rate is 15% � 0.15.

c. What is the decay factor in this situation?

The decay factor is 0.85.

d. Use the equation from part a to estimate the value of your car 5 yearsfrom now.

e. Use the trace and table features of your graphing calculator to check yourresults in part d.

V 15 2 � 20,00010.85 25 � 8874.11

V 1t 2 � 20,00010.85 2t

P 170 2 � 120.611.0125 270 � 287.7

P(t) = 120.6 • 1.0125t

300

100

200

100 20 30 40 50 60 70t

Number of Years since 1930U

.S. P

op

ula

tio

n(i

n m

illio

ns)

P


f. Use the trace or table features of your graphing calculator to determinewhen your car will be worth $10,000.

The value will be $10,000 when years.t � 4.3

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 316

Most photocopy machines allow for enlarging or reducing the size of the original.Suppose you have a chart you wish to photocopy for a report you need to submitto your supervisor. The chart is 10 inches wide by 7 inches high. To reduce thesize of the copy 20%, you set the machine to reduce by taking 80% of the originaldimensions.

1. What will be the dimensions of your photocopy?8 in. wide and 5.6 in. high

2. What is the percentage reduction in area of the photocopy?original area: in.2; new area: in.2; new area is

, or 64% of the original. So the area was reduced by 36%.

3. If your chart must fit in a space only 4 inches high in your report, how manytimes would you need to reduce the original? (Assume the photocopier is setat 80% reduction.)3 times

4. a. Complete the following table showing the dimensions of the chart after x80% reductions (photocopies of photocopies). Record each length to thenearest hundredth of an inch.

44.8 � 70 � 0.648 # 5.6 � 44.810 # 7 � 70

Project Activity 3.4 Photocopying Machines 317

PROJECTACTIVITY 3.4

PhotocopyingMachines

OBJECTIVES

1. Generate data given thegrowth or decay rate ofan exponential function.

2. Write exponentialfunctions given thegrowth or decay rate.

3. Graph exponentialfunctions from data.

4. Determine doubling andhalving times fromexponential functions.

x, THE NUMBER OF 0 1 2 3 4 5 6 7 8 9 1080% REDUCTIONS

h 11x22, HEIGHT (in.) 7 5.6 4.48 3.58 2.87 2.29 1.84 1.47 1.17 0.94 0.75

w 11x22, WIDTH (in.) 10 8 6.4 5.12 4.10 3.28 2.62 2.10 1.68 1.34 1.07

Smaller and Smaller

b. If you were actually to perform ten 80% reductions (photocopies ofphotocopies), what do you think your results would look like?

Copies of copies deteriorate in quality.

5. Use the data from your table to plot points for two curves on separate axes.Use the number of reductions, x, versus the heights of the reduced copies forone curve, and the number of reductions versus the widths of the reducedcopies for a second curve. Describe how the two graphs are similar and howthey are different.

Each graph represents a decreasing exponential function. The widthfunction starts at a larger value (10 rather than 7).

6

8

10

2

4

20 4 6 8 10x

Number of 80% Reductions

Wid

th (

in.)

w(x)

6

8

10

2

4

20 4 6 8 10Number of 80% Reductions

Hei

gh

t (i

n.)

h(x)

DANFMC03_0321447808 pp3 1/6/07 11:23 AM Page 317

6. a. Notice that each entry in the table for h 1x2 or w 1x2 can be obtained by mul-tiplying the previous entry by a constant factor. What is the constant fac-tor in this situation?

0.80

b. What is the practical meaning of the constant factor in this situation?

Each photocopy is 80% of the previous one.

7. Determine the equations that define the functions h and w, where x is thenumber of 80% reductions. Enter these functions on your grapher to verifyyour work in Problems 4 and 5.

width:

height:

8. The 80% reduction resulted in a decreasing function. What kind of percentagewould you need to enter on the photocopier to result in an increasing function?

a percentage greater than 100%

9. a. Complete the following table showing the width of a 10-inch-wide chartin which x represents the number of 20% enlargements.

h 1x 2 � 710.80 2xw 1x 2 � 1010.80 2x


x, NUMBER OF 20% ENLARGEMENTS 0 1 2 3 4 5

w11x22, WIDTH (in.) 10 12 14.4 17.3 20.7 24.9

b. What is the growth factor? What is the practical meaning of the growthfactor in this situation?

The growth factor is 1.20. Each enlargement is 120% of the previousversion.

c. Write an equation for w in terms of x, where x is the number of 20%enlargements.

d. Use your graphing calculator to graph the function.

w(x) = 10(1.2)x15

20

5

10

10 2 3 4 5x

Number of 20% Enlargements

Wid

th (

in.)

w(x)

w 1x 2 � 1011.20 2x

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 318

10. a. If the machine you are using will enlarge only at 20%, how many timeswould you need to copy the 10-inch width to make it at least 20 incheswide?

4 times

b. Will the chart ever be exactly 20 inches wide? Explain.

No; at four enlargements, the width will be 20.736 inches.

c. Describe in detail how you found your answers.

I used the table feature of the calculator.

11. Assuming a constant 20% enlargement, how many copies would it take to getyour original 10-inch width to grow to at least 40 inches? At least 80 inches?

12. Complete the following table with your results from Problems 10 and 11.

80 in. S 12 copies

40 in. S 8 copies

Project Activity 3.4 Photocopying Machines 319

x, NUMBER OF 20% ENLARGEMENTS 0 4 8 12

WIDTH (in.) 10 20.7 43.0 89.2

13. Use your graphing calculator to determine how many copies are needed todouble the output.

4 copies

14. Examine the height and width functions given in Problem 7,and

a. Complete the following table.

w1x 2 � 1010.8 2x.h1x 2 � 710.8 2x

x, NUMBER OF 80% 0 3 6 9REDUCTIONS

h11x22, HEIGHT (in.) 7 3.6 1.84 0.940

x, NUMBER OF 80% 0 3 6 9REDUCTIONS

w 11x22, WIDTH (in.) 10 5.12 2.62 1.34

b. What is the half-life of each function?

The half-life of each function is approximately x � 3.

DANFMC03_0321447808 pp3 1/6/07 11:24 AM Page 319

Congratulations, you have inherited $20,000! Your grandparents suggest that youuse half of the inheritance to start a retirement fund. Your grandfather claims thatan investment of $10,000 could grow to over half a million dollars by the time ofretirement. You are intrigued by this statement and decide to investigate if thiscould possibly happen.

Suppose you deposit $10,000 in the bank at a 6.5% annual interest rate. After1 year, your balance is

.

The interest, $650, earned during the year becomes part of the new balance. At theend of the second year, your balance is

.

Note that you made interest on the original deposit, plus interest on the first year’sinterest. In this situation, we say that interest is compounded. Usually, the com-pounding occurs at fixed intervals (typically at the end of every year, quarter,month, or day). In the preceding situation, interest is compounded annually.

10,650 � 0.065110,650 2 � 10,650 � 692.25 � 11,342.25

10,000 � 0.065110,000 2 � 10,000 � 650 � 10,650


ACTIVITY 3.5

Compound Interest

OBJECTIVE

1. Apply the compoundinterest and continuouscompounding formulas toa given situation.

This formula is referred to as the compound interest formula.

Example 1 You invest $100 at 4% compounded quarterly. How much moneydo you have after 5 years?

SOLUTION

The principal is $100, so The annual interest rate is 4%, so Interest is compounded quarterly, that is, 4 times per year, so The moneyis invested for 5 years, so Substituting the values for the P, r, n, and t in thecompound interest formula, you have

A � 100 11 �0.04

4 24 # 5 � $122.02.

t � 5.n � 4.

r � 0.04.P � 100.

1. a. Suppose you deposit $10,000 in an account that has a 6.5% annual interestrate (usually referred to as APR, for annual percentage rate), and whoseinterest is compounded annually Substitute the appropriate valuesfor P, n, and r into the compound interest formula to get the balance, A, as afunction of time, t.

A � 10,000 # 11 �0.065

1 21 # t

1n � 1 2.

If interest is compounded, then the current balance is given by the formula

where A is the current amount, or balance, in the account;

P is the principal (the original amount deposited);

r is the annual interest rate (annual percentage rate in decimal form);

n is the number of times per year that interest is compounded; and

t is the time in years the money has been invested.

A � P ̌11 �rn2nt,

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 320

b. Use the compound interest formula from part a to determine your balance,A, at the end of the first year

, or $10,650.00

c. What will be the amount of interest earned in the first year?

$650

d. Use the compound interest formula developed in part a and complete thefollowing table.

A � 10,00011 �0.065

1 21 # 1 � 10,650

1t � 1 2.

Activity 3.5 Compound Interest 321

t, YEAR 0 1 2 3 4

A, BALANCE 10,000.00 10,650.00 11,342.25 12,079.50 12,864.66

e. The compound interest formula in part a defines A as an exponentialfunction of t. Identify the base.

1.065

f. Is the base a growth or decay factor? Explain.

Because the base is greater than 1, it is a growth factor.

2. a. Suppose you deposit the $10,000 into an account that has the same interestrate (APR) of 6.5%, with compounding quarterly rather thanannually Write a new formula for your balance, A, as a functionof time.

b. What would be your balance after the first year?

, or $10,666.02

c. Use the table feature of your calculator to determine the balance at theend of each year for 10 years, and record the values in the table inProblem 3 under (compounded quarterly).

d. What is the base of this exponential function?

3. Now deposit your $10,000 into a 6.5% APR account with monthly compound-ing and then in an account with daily compounding Useyour graphing calculator and the appropriate formula to complete the followingtable.

1n � 365 2.1n � 12 2

11 �0.065

4 24 � 11.01625 24 � 1.0666

n � 4

A � 10,00011 �0.065

4 24 # 1 � 10,666.02

A � 10,00011 �0.065

4 24t

1n � 1 2. 1n � 4 2

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 321

4. In Problem 3, you calculated the balance on a deposit of $10,000 at an annualinterest rate of 6.5% that was compounded at different intervals. After 10 years,which account has the higher balance? Does this seem reasonable? Explain.

Compounding daily results in the largest balance after 10 years. Thebalance is greater where the number of compounding periods per yearis larger, since you compute interest on interest more frequently.

Continuous Compounding

You could extend this problem so that interest is compounded every hour or everyminute or even every second. However, compounding more frequently than everyhour does not increase the balance very much.

To discover why this happens, take a closer look at the exponential functions fromProblems 1–3.

Can you discover a pattern in the form of the underlined expressions?

A � 10,000 11 �0.065365 2365

# t � 10,000 c 11 �

0.065365 2365 d

t

A � 10,000 11 �0.065

12 212 # t � 10,000 c 11 �

0.06512 212 d

t

A � 10,000 11 �0.065

4 24 # t � 10,000 c 11 �

0.0654 24 d t

A � 10,000 11 �0.065

1 221 # t � 10,000 311 � 0.065 21 4 t


0 10,000.00 10,000.00 10,000.00

1 10,666.02 10,669.72 10,671.53

2 11,376.39 11,384.29 11,388.15

3 12,134.08 12,146.72 12,152.90

4 12,942.22 12,960.20 12,969.00

5 13,804.20 13,828.17 13,839.91

6 14,723.58 14,754.27 14,769.30

7 15,704.19 15,742.39 15,761.10

8 16,750.12 16,796.69 16,819.50

9 17,865.70 17,921.60 17,948.97

10 19,055.59 19,121.84 19,154.30

COMPARISON OF $10,000 PRINCIPAL IN 6.5% APR ACCOUNTSWITH VARYING COMPOUNDING PERIODS

t n � 365n � 12n � 4

Each formula can be expressed as where forand 365. The number b is called the growth factor, and n is the

number of compounding periods per year.n � 1, 4, 12,

b � 11 �0.065

n 2nA � 10,000b t,

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 322

Example 2 If in the formula thenThe number 1.06660 is the growth

factor.b � (1 � 0.065

4 )4 � 1.06660.b � (1 � 0.065

n )n,n � 4


5. Determine the value of b in the following table, where Round to five decimal places.

b � 11 �0.065

n 2n.

n, NUMBER OF COMPOUNDING PERIODS 1 4 12 365

b, GROWTH FACTOR 1.065 1.06660 1.06697 1.06715

The growth rate is the percentage by which the balance grows by in 1 year. It iscalled the effective yield, . Notice that as the number of compounding periodsincreases, the effective yield increases. This means that with the same annual inter-est rate (APR), your investment will earn more with more compounding periods.

re

PROCEDURE

To calculate the effective yield

1. Determine the growth factor .

2. Subtract 1 from b and write the result as a decimal.

re � b � 1 � a1 �rnbn

� 1

b � a1 �rnbn

Example 3 Determine the effective yield for an annual percentage rate (APR)of 4.5% compounded monthly.

SOLUTION

r � 4.5% � 0.045, n � 12

re � 4.594%

re � b � 1 � a1 �rnbn

� 1 � a1 �0.045

12b12

� 1 � 1.04594 � 1 � 0.04594

6. a. If interest is compounded hourly, then Computethe growth factor, b, for compounding hourly, using an APR of 6.5%.

b. Determine the effective yield associated with each of the growth factors inthe following table.

b � 11 �.065876028760 � 1.06716

n � 365 # 24 � 8760.

n 1 4 12 365

GROWTH FACTOR, b 1.065 1.0666 1.06697 1.06715

EFFECTIVE YIELD re 6.5% 6.66% 6.697% 6.715%

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 323

c. Write a sentence comparing the growth factor b for compounding hourly,to that for daily compounding,

The growth factors differ by about 0.00001. The extra compoundinghad little effect.

If the compounding periods become shorter and shorter (compounding every hour,every minute, every second), n gets larger and larger. If you consider the period tobe so short that it’s essentially an instant in time, you have what is called continuouscompounding. Some banks use this method for compounding interest.

The compound interest formula is no longer used when interestis compounded continuously. The following develops a formula for continuouscompounding.

Step 1. Rewrite the given formula as indicated using properties of exponents.

since

Step 2. Let It follows that Note that as n gets very large, the valueof x also gets very large.

Step 3. Substituting x for and for in the rewritten formula in step 1, you have

7. a. Now take a closer look at the expression Enter intoyour calculator as a function of x. Display a table that starts at 0 and isincremented by 100. The results are displayed below.

b. In the table of values, why is there an error at ?

There is an error at because is undefined.

c. Scroll down in the table and describe what happens to the output,as the input, x, gets very large.

As x gets very large does not change very much.11 �1x 2x

11 �1x 2x,

10x � 0

x � 0

11 �1x 2x11 �

1x 2x.

A � P c 11 �rn2n>r d rt

� P c 11 �1x 2x d rt

.

rn,1

xnr

rn �

1x.

nr � x.

nr

# rt � ntA � P 11 �rn2nt � P c 11 �

rn2n>r d rt

,

A � P11 �rn2nt

n � 365.n � 8760,


The letter e is used to represent the number that approaches as x getsvery large. This notation was devised by mathematician Leonhard Euler(1707–1783). Euler used the letter e to denote this number. The number isirrational and its decimal representation never ends and never repeats.

11 �1x 2x

8. The number e is a very important number in mathematics. Find it on yourcalculator, and write its decimal approximation below. How does this approx-imation compare to the result in Problem 7a?

; very closee � 2.718281828

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 324

You are now ready to complete the compound interest formula for continuous

compounding. Substituting e for in you obtain the

continuous compounding formula

A � P c 11 �1x 2x d rt

,11 �1x 2x


Example 4 You invest $100 at a rate of 4% compounded continuously. Howmuch money will you have after 5 years?

SOLUTION

The principal is $100, so The annual interest rate is 4%, so The money is invested for 5 years, so Because interest is compoundedcontinuously, you use the formula for continuous compounding as follows.

A � 100e0.04 # 5 � $122.14

t � 5.r � 0.04.P � 100.

9. a. Calculate the balance of your $10,000 investment in 10 years with anannual interest rate of 6.5% compounded continuously.

The formula used for the preceding result was . Comparingwith shows that the growth factor is .

b. Determine the growth factor in this situation.

b � 1.06716

c. What is the effective yield of an annual interest rate of 6.5% compoundedcontinuously?

� b � 1 � 1.06716 � 1 � 0.06716 � 6.716%

10. a. Historically, investments in the stock market have yielded an average rateof 11.7% per year. Suppose you invest $10,000 in an account at an 11%annual interest rate that compounds continuously. Use the formula

to determine the balance after 35 years.

A � 10,000e0.11 # 35 � $469,930.63

A � Pert

re

b � e0.065A � 10,000btA � 10,000e0.065tA � 10,000e0.065t

A � 10,000e.065 # 10 � $19,155.41

where A is the current amount, or balance, in the account;

P is the principal;

r is the annual interest rate (annual percentage rate in decimal form);

t is the time in years that your money has been invested; and

e is the base of the continuously compounded exponential function.

A � Pert,

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 325

b. What is the balance after 40 years?

c. Your grandfather claimed that $10,000 could grow to more than half amillion dollars by retirement time (40 years). Is your grandfather correctin his claim?

yes

A � 10,000e0.11 # 40 � $814,508.69


1. The formula for compounding interest is

2. The formula for continuous compounding is

3. If the number of compounding periods is large,A � P 11 �

rn2nt is approximated by A � Pert.

A � Pert.

A � P 11 �rn2nt.SUMMARY

ACTIVITY 3.5


1. You inherit $25,000 and deposit it into an account that earns 4.5% annual interestcompounded quarterly.

a. Write an equation that gives the amount of money in the account aftert years.

b. How much money will be in the account after 10 years?

c. You want to have approximately $65,000 in the bank when your first childbegins college. Use your graphing calculator to determine in how manyyears you will reach this goal.

approximately 21.4 yr.

d. If the interest were to be compounded continuously at 4.5%, how muchmoney would be in the account after 10 years?

e. Use your graphing calculator to determine in how many years you wouldreach your goal of $65,000, if the interest is compounded continuously.


f. Should you look for an investment account that will be compoundedcontinuously?

Yes.

A � 25,000e0.045 # 10 � $39,207.80

A � 25,00011 �0.045

4 24 # 10 � $39,109.42

A � 25,00011 �0.045

4 24t


DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 326

2. You deposit $2000 in an account that earns 5% annual interest compoundedmonthly.

a. What will be your balance after 2 years?

b. Estimate how long it would take for your investment to double.


c. Identify the annual growth rate and the growth factor.

The growth factor is The growth rate is0.0512, or 5.12%.

3. Your friend deposits $1900 in an account that earns 6% compoundedcontinuously.

a. What will be her balance after 2 years?

b. Estimate how long it will take for your friend’s investment to double.


4. You are 25 years old and begin to work for a large company that offers youtwo different retirement options.

Option 1. You will be paid a lump sum of $20,000 for each year you workfor the company.

Option 2. The company will deposit $10,000 into an account that will payyou 12% annual interest compounded monthly. When youretire, the money will be given to you.

Let A represent the amount of money you will have for retirement aftert years.

a. Write an equation that represents option 1.

b. Write an equation that represents option 2.

c. Use your graphing calculator to sketch a graph of the two options on thesame axis.

Xmin � 0, Xmax � 45, Ymin � 0, Ymax � 1,300,000

d. If you plan to retire at age 65, which would be the better plan? Explain.

After 40 years, option 1 will yield $800,000.00 and option 2 will yield$1,186,477.25.

A � 10,00011 �0.1212 212

# t

A � 20,000t

A � 1900e0.06 # 2 � $2142.24

b � 11 �0.0512 212 � 1.0512.

A � 200011 �0.0512 212

# 2 � $2209.88


DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 327

e. If you decide to retire at age 55, which would be the better plan? Explain.

After 30 years, option 1 will yield $600,000.00 and option 2 will yield$359,496.41.

f. Use your graphing calculator to determine at what age it would not makea difference which plan you choose.

If you work 36 or more years, you should choose option 2.

5. The compound interest formula that gives the balance in an account with aprincipal of $1500 that earns interest at the rate of 4.8% compounded monthly

is . Compare this formula to the exponential

equation .

a. What takes the place of b in the compound interest formula?

b. What is the value of the growth factor b?

b � 1.04907

c. What is the effective rate?

re � 4.907%

6. The compound interest formula that gives the balance in an account witha principal of $1500 that earns interest at the rate of 4.8% compoundedcontinuously is . Compare this formula to the exponentialequation .

a. What takes the place of b in the compound interest formula?

b � e0.048

b. What is the value of the growth factor b?

b � 1.04917

c. What is the effective rate?

re � 4.917%

A � 1500 # btA � 1500 # e0.048t

b � a1 �0 .048

12b12

A � 1500 # bt

A � 1500 # a1 �0.048

12b12t


DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 328

The U.S. Bureau of the Census reported that the U.S. population on April 1, 2000,was 281,421,906. The U.S. population on April 1, 2001, was 284,236,125.

1. Assuming exponential growth, the U.S. population y can be modeled by theequation y � abt, where t is the number of years since April 1, 2000. Therefore,t � 0 corresponds to April 1, 2000.

a. What is the initial value, a?

a � 281,421,906

b. Determine the annual growth factor, b, for the U.S. population.

b � 284,236,125 � 281,421,906 � 1.01

c. What is the annual growth rate?

r � b � 1 � 1.01 � 1 � 0.01 � 1%

d. Write the equation for the U.S. population as a function of t.

y � 281,421,90611.012tThe U.S. population did not remain constant at 281,421,906 from April 1, 2000, toMarch 31, 2001, and then jump to 284,236,125 on April 1, 2001. The populationgrew continuously throughout the year. The exponential function used to modelcontinuous growth is the same function used to model continuous compounding foran investment.

Recall from Activity 3.5 that the formula for continuous compounding is ,where A (output) is the amount of the investment, P is the initial principal, r is thecompounding rate, t (input) is time, and e is the constant irrational number. Whenthis function is used more generally, it is written as , where

A has been replaced by y, the output;

P replaced by a, the initial value; and

r replaced by k, the continuous growth rate.

Now, the exponential growth model for the U.S. population determined inProblem 1d, written in the form y � abt, can be rewritten equivalently in the con-tinuous growth form, y � aekt. Since y represents the same output value in eachcase, abt � aekt. Since a represents the same initial value in each model, it followsthat bt � ekt.

2. a. Notice that can be written as . How are b and ek related?

b � ek

b. Set the value of b determined in Problem 1 1b � 1.012 equal to ek andsolve for k, the continuous growth rate. Solve the equation graphicallyby entering Y1 � ex and Y2 � 1.01. Use the window Xmin � 0,Xmax � 0.02, Ymin � 1, and Ymax � 1.02.

k � 0.00995

c. Rewrite the U.S. population function from Problem 11y � 281,421,90611.012 t 2 in the form

y � 281,421,906e0.00995t

y � a # e kt.

1e k 2te

kt

y � aekt

A � Pert

Activity 3.6 Continuous Growth and Decay 329

ACTIVITY 3.6

Continuous Growthand Decay

Objectives

1. Discover the relationshipbetween the equations ofexponential functionsdefined by y � abt and theequations of continuousgrowth and decayexponential functionsdefined by y � aekt.

2. Solve problems involvingcontinuous growth anddecay models.

3. Graph base e exponentialfunctions.

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 329

Notice that an annual growth rate of r � 0.01 � 1% is equivalent to a continuousgrowth rate of k � 0.00995 � 0.995%.


Whenever growth is continuous at a constant rate, the exponential model used todescribe it is , where k is the constant continuous growth rate, a is theamount present initially (when t � 0) and e is the constant irrational numberapproximately equal to 2.718.

y � aekt

Example 1

a. Rewrite the equation into a continuous growth equation ofthe form .

SOLUTION

, k � 0.207,

b. What is the continuous growth rate?

SOLUTION

0.207 � 20.7%

c. What is the initial amount present (when t � 0)?

SOLUTION

42

y � 42e0.207t1.23 � e k

y � ae kt

y � 4211.23 2t

Now, consider a situation which involves continuous decay at a constant percentagerate. Tylenol (acetaminophen) is metabolized in your body and eliminated at the rateof 24% per hour. You take two Tylenol tablets (1000 milligrams) at 12 noon.

3. Assume that the amount of Tylenol in your body can be modeled by an expo-nential function , where t is the number of hours from 12 noon.

a. What is the initial value, a, in this situation?

a � 1000

b. Determine the decay factor, b, for the amount of Tylenol in your body.

b � 1 � r � 1 � 0.24 � 0.76

c. Write an exponential equation for the amount of Tylenol in your body as afunction of t.

Q(t) � 100010.762tOf course, the amount of Tylenol in your body does not decrease suddenly by24% at the end of each hour; it is metabolized and eliminated continuously. Theequation can also be used to model a quantity that decreases at a contin-uous rate.

Recall in Problem 2a, you compared to and established that .b � ek

y � aekty � abt

y � aekt

Q � abt

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 330

4. a. The value of b for the Tylenol equation is 0.76. Set b � 0.76 equal to ek andsolve for k graphically as in Problem 2b. Use the window Xmin � �1,Xmax � 0, Ymin � 0, and Ymax � 1.

k � �0.274

b. Write the equation for the amount of Tylenol in your body, Q � 100010.762t,in the form .

Q � 1000e�0.274t

Notice that the value of k in Problem 4 is negative. Whenever 0 b 1, then b isa decay factor and the value of k will be negative. A decreasing exponential func-tion written in the form will have k < 0, and is the continuous rate ofdecrease.

0k 0y � aekt

Q � aekt


For exponential decrease (decay) at a continuous constant rate, the model is used, where k < 0, is the constant continuous decrease (decay) rate, a is theamount present initially, when t � 0, and e is the constant irrational numberapproximately equal to 2.718.

0k 0 y � aekt

Example 2

a. Rewrite the decay equation as a function of the form.

SOLUTION

, k � �0.186,

b. What is the continuous percentage rate of decay?

SOLUTION

0.186 � 18.6%

c. What is the initial amount present when t � 0?

SOLUTION

12.5

y � 12.5e�0.186t0.83 � ek

y � aekty � 12.510.83 2t

Graphs of Exponential Functions

5. The graph of an increasing exponential function has the shape representedbelow.

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 331

a. If the equation for the preceding graph is written as , what do youknow about the values of a and b?

a 0, b 1

b. If the equation for the preceding graph is written as , what doyou know about the values of a and k?

a 0, k 0

6. The graph of a decreasing exponential function has the shape representedbelow.

a. If the equation for the preceding graph is written as , what do youknow about the values of a and b?

a 0, 0 b 1

b. If the equation for the preceding graph is written as , what do youknow about the values of a and k?

a 0, k 0

7. Identify the given exponential function as increasing or decreasing. In eachcase give the initial value and rate of increase or decrease.

a.The initial value is 2500 and P is increasing at the continuous rateof 4%.

b.The initial value is 400 and Q is decreasing at the rate of 14%.

c.The initial value is 75 and A is increasing at the rate of 3.2%.

d.The initial value is 12 and R is decreasing at the continuous rateof 12%.

R � 12e�0.12t

A � 7511.032 2t

Q � 40010.86 2t

P � 2500e0.04t

y � aekx

y � abx

y � ae kx

y � ab x


DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 332


1. In 2004, Charlotte, North Carolina, was ranked twentieth in population of all theU.S. cities. The population of Charlotte has been increasing steadily and the ex-ponential function defined by models this population from1990 to 2004, where x represents the number of years since 1990 and f 1x2 repre-sents the total population in thousands. (Source: U.S. Bureau of the Census.)

a. Is this an increasing or decreasing exponential function? Explain.

The function is increasing because in the model b � 1.

f 1x 2 � 397.411.03 2x

1. When a quantity increases or decreases continuously at a constant rate, theamount present at time t can be modeled by , where a is the initialquantity at t � 0. If the quantity is increasing, then k � 0 and k is thecontinuous rate of increase. If the quantity is decreasing, then k < 0 and is the continuous rate of decrease.

2. The graph of an increasing exponential function of the form, where a � 0 and b � 1 or k � 0 will be shaped like

3. The graph of a decreasing exponential function of the form or, where a � 0 and 0 � b � 1 or k � 0 will be shaped like

Notation

The function may be written in other forms, such as , where

the initial value is (y sub zero) or , where the initial value is (Q sub zero).

Q0Q � Q0ekty0

y � y0 ekty � ae

kt

y

x

y � ae kt

y � ab x

y

x

y � abx or y � aekt

0k 0

y � ae kt

SUMMARYACTIVITY 3.6



DANFMC03_0321447808 pp3 1/6/07 1:21 PM Page 333

b. Determine the annual growth or decreasing (decay) rate from the model.

It is a growth rate: r � b � 1, r � 1.03 � 1 � 0.03 � 3%.

c. According to the model, what is the initial value? What does the initialvalue mean in this situation?

The initial value is 397.4 thousand. According to the model, it is thepopulation in thousands of Charlotte, North Carolina, in 1990.

d. The equation for continuous growth is . Set the value of b in yourmodel equal to ek, and use your graphing calculator to determine thevalue for k graphically. This will be the continuous growth or decay rate.

k � 0.0296

e. Rewrite the population function in the form of .

f. Use the function from part e and predict the population of Charlotte in2010.

The population of Charlotte will be 718.3 thousand in 2010 if thegrowth continues at the same rate.

2. The table shows the smoking prevalence among U.S. male adults (18 yearsand over) as a percent of the population.

y � 397.4e0.0296t

y � aekt

y � aekt


Where There’s SmokeSMOKING PREVALENCE AMONG U.S. MALE ADULTS

YEAR 1955 1965 1970 1980 1990 2000 2002

YEARS SINCE 1955 0 10 15 25 35 45 47

% OF POPULATION 56.9 51.9 44.1 37.6 28.4 25.7 25.2

Source: The U.S. Center for Disease Control and Prevention.

This data can be modeled by an exponential function , where tequals the number of years since 1955 and p is the smoking prevalence amongmale adults as a percent of the U.S. population.

a. Is this an increasing or decreasing exponential function? Explain.

The function is decreasing because as the years increase, the percentis decreasing. In the model, b is 0 b 1.

p1t 2 � 58.810.98 2t

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 334

b. Sketch a graph of the function using the data in the table. Does the graphreinforce your answer in part a?

yes

c. Determine the growth or decreasing (decay) rate from the model.

It is a decreasing (decay) rate: 1 � 0.98 � 0.02, or 2%.

d. According to the model, what is the initial value?

The initial value is 58.9% from the model. It differs from the tablevalue because the model only approximates the given data.

e. The equation for the continuous growth or decay is . Set thevalue of b in your model equal to ek, and use your graphing calculator todetermine the value for k graphically. This will be the continuous growthor decay rate.

k � �0.0202, the continuous decay rate is 0.0202

f. Rewrite the smoking function in the form of .

g. Use the function from part f and determine the percent of males in theU.S. population that will be smoking in 2006.

Approximately 21% of males in the U.S. population will be smokers in2006 if the decline continues at the same rate.

3. Identify the given exponential function as increasing or decreasing. In eachcase give the initial value and rate of increase or decrease.

a.The initial value is 33 and R is increasing at the rate of 9.7%.

R1t 2 � 3311.097 2t

y � 58.9e�.0202t

y � ae kt

y � ae kt

0

25

30

35

40

45

50

55

60

10 20

Sm

oki

ng

Pre

vale

nce

Am

on

g U

.S.

Ad

ult

Mal

es

Years since 195530 40 50

y

x


DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 335

b.The initial value is 97.8 and f (x) is decreasing continuously at therate of 23%.

c.The initial value is 3250 and S is decreasing continuously at the rateof 27%.

d.The initial value is 0.987 and B is increasing continuously at the rateof 7.6%.

4. Sketch a graph of each of the following and verify using your graphingcalculator.

a.

b.

30

60

90

120

10

F(x)

x

f 1x 2 � 10e�0.3x

80

160

240

320

400

480

20 30 4010

y

x

y � 20e0.08x

B � 0.987e0.076t

S � 325010.73 2t

f 1x 2 � 97.8e�0.23x


DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 336

c. Compare the two graphs. Include shape, direction, and initial values.

Both graphs are exponential curves. The first is increasing. The secondis decreasing. The initial value for the first is 20. The initial value forthe second is 10.

5. Strontium 90 is a radioactive material that decays according to the functiondefined by where is the amount present initially and t istime in years.

a. If there are 20 grams of strontium 90 present today, how much will bepresent in 20 years?

b. Use the graph of the function to approximate how long it will take for 20 grams to decay to 10 grams, 10 grams to decay to 5 grams, and5 grams to decay to 2.5 grams. The length of time is called the half-life.In general, a half-life is the time required for half of a radioactive sub-stance to decay.

The half-life is approximately 28.4 years.

c. Identify the annual decay rate and the decay factor.

The decay factor is The decay rate is 0.0241, or2.41%.

6. When drugs are administered into the bloodstream, the amount present de-creases continuously at a constant rate. The amount of a certain drug in thebloodstream is modeled by the function where is theamount of the drug injected (in milligrams) and t is time (in hours).

a. Suppose that 10 milligrams are injected at 10:30 A.M., how much of thedrug is still in the bloodstream at 2:00 P.M.?

At 2:00 P.M. t � 3.5, so milligrams.

b. If another dose needs to be administered when there is 1 milligram of thedrug present in the bloodstream, approximately when should the nextdose be given (to the nearest quarter hour)?

In about 6.5 hours at about 5:00 P.M. the next dose should be given.

7. The amount of credit-card spending from Thanksgiving to Christmas hasincreased by 14% per year since 1987. The amount, A, in billions of dollarsof credit-card spending during the holiday period in a given year can bemodeled by

where x represents the number of years since 1987.

a. How much was spent using credit cards from Thanksgiving to Christmasin 1996?

In 1996 so billion dollars.A � 36.2e0.14 # 9 � 127.62x � 9,

A � f 1x 2 � 36.2e0.14x,

y � 10e�0.35 # 0.35 � 2.94

y0y � y0e�0.35˛t,

b � e�0.0244 � 0.9759.

y � 20e�0.0244 # 20 � 12.277

y0y � y0e�0.0244˛t,


DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 337

b. Sketch a graph of the credit-card function.

c. What is the vertical intercept of the graph? What is the practical meaningof the intercept in this situation?

The vertical intercept is 10, 36.22. There was $36,200,000,000 in holidaycredit-card spending in 1987.

d. Determine, graphically and numerically, the year when credit-card spend-ing reached 75 billion dollars.

1993 1x � 52e. What is the doubling time?


8. E. coli bacteria are capable of very rapid growth, doubling in number ap-proximately every 49.5 minutes. The number, N, of E. coli bacteria per mil-liliter after x minutes can be modeled by the equation

a. What is the initial number of bacteria per milliliter?

500,000

b. How many E. coli bacteria would you expect after 99 minutes? (Hint:There will be two doublings.) Verify your estimate using the equation.

500,000 doubled twice is 2,000,000 per milliliter.

c. Use a graphing or numerical approach to determine the elapsed timewhen there would be 20,000,000 E. coli bacteria per milliliter.

There will be 20,000,000 bacteria when minutes, or 4 hours24 minutes.

x � 264

N � 500,000e0.014x.

A = 36.2 • e 0.14x

90

120

30

60

20 4 6 8 10x

Years after 1987

Cre

dit

Car

d S

pen

din

g(i

n b

illio

ns)

A


DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 338

Activity 3.7 Bird Flu 339

ACTIVITY 3.7

Bird Flu

Objectives

1. Determine the regressionequation of an exponen-tial function that best fitsthe given data.

2. Make predictions usingan exponential regressionequation.

3. Determine whether a lin-ear or exponential modelbest fits the data.

In 2005, the Avian Flu, also known as Bird Flu, received international attention.Although there were very few documented cases of the Avian Flu infecting hu-mans worldwide, world health organizations including the Centers for DiseasesControl in Atlanta expressed concern that a mutant strain of the Bird Flu virus thatcould infect humans via human-to-human contact would develop and produce aworldwide pandemic.

The infection rates, the number of people each infected person will infect, and theincubation period, the time between exposure and the development of symptomsof this flu, cannot be known exactly. However, this information can be approxi-mated by studying the infection rates and incubation periods of existing strains ofthe virus.

A very conservative infection rate would be 1.5 and a reasonable incubation pe-riod would be about 15 days. This means that the first infected person could beexpected to infect 1.5 people in roughly a half of a month. This assumes that thespread of the virus is not checked by inoculation or vaccination.

So the total number of infected people 0.5 months after the first person wasinfected would be 2.5, the sum of the original infected person and the 1.5 newlyinfected people. During the second half month, the 1.5 newly infected peoplewould infect 1.5 � 1.5 � 2.25 new people. This means you have 2.25 people toadd to the 2.5 people previously infected, or approximately 5 people infected withBird Flu at the end of the first month.

1. The following table represents the total number of people who could be in-fected with a mutant strain of Bird Flu over a period of 5 months. Completethe table. Round each value to the nearest whole person.

MONTHS SINCE THEFIRST PERSON WAS 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5INFECTED

NUMBER OF NEWLY 1.5 2.25 3.375 5 8 12 18 27 41 61INFECTED

TOTAL NUMBER OF 1 2.5 5 8 13 21 33 51 78 119 180PEOPLE INFECTED

The Spread of Bird Flu

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 339

2. Let t represent the number of months since the first person was infected andT represent the total number of people infected with the Bird Flu virus. Cre-ate a scatterplot of the data below.

3. Does the scatterplot indicate a linear relationship between t and T? Explain.

No. The relationship does not appear to be linear since the data pointsdo not lie on a straight line.

4. a. Use your calculator to model the data with an exponential function. Useoption 0:ExpReg in the STAT CALC menu to determine an exponentialfunction that best fits the given data. Record the regression equation of theexponential model below. Round a and b to the nearest 0.001.

Y � T � 1.55112.6842tb. Sketch a graph of the exponential model using your calculator and add it

to the scatterplot in Problem 2.

c. What is the practical domain of this function?

t � 0 or until everyone is infected or until the virus is checked byinoculation or vaccination.

d. What is the y-intercept of the graph? How does it compare to the actualinitial value 1t � 02 from the table?

10, 1.5512; It is a little high, but pretty close to 10, 12.5. a. Use the exponential model to determine the total number of infected

people 1 year after the initial infection 1t � 122 provided the virus isunchecked. Round your result to the nearest whole person.

T 1122 � 216,771

b. Use the exponential function to write an equation that can be used to de-termine when the virus will first infect 2,000,000 people.

2,000,000 � 1.55112.6842t

0

50

100

150

200

1 2

Tota

l Nu

mb

er o

f P

eop

le In

fect

ed

Months Since First Person Was Infected3

t

T

4 5 6


DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 340

c. Solve the equation in part b using a graphing approach. Use the intersectfeature of your calculator; the screen containing the solution shouldresemble the following.

d. Interpret the meaning of your solution in part c.

The virus will infect its 2,000,000 person in the 15th month after it hasinfected its first person.

College Graduates

According to the U.S. Department of Education, the number of college graduatesincreased significantly during the twentieth century. The following table gives thenumber (in thousands) of college degrees awarded from 1900 through 2000.


1900 30

1910 54

1920 73

1930 127

1940 223

1950 432

1960 530

1970 878

1980 935

1990 1017

2000 1180

YEAR NUMBER OF COLLEGEGRADUATES (thousands)

College Bound

6. Let t represent the number of years since 1900 (t � 0 corresponds to 1900,t � 10 to 1910, etc.). Let N represent the number of college graduates (inthousands) at time t. Sketch a scatterplot of the given data on your graphingcalculator. Your scatterplot should appear as follows.

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 341

7. a. Use your graphing calculator to determine the regression equation of anexponential function that best fits the given data.

b. Sketch a graph of the exponential model using your graphing calculator.

c. What is the practical domain of this exponential function?

0 to 100

d. What is the vertical intercept of the graph? How does it compare to theactual initial value 1t � 02 from the table?

The vertical intercept is 10, 40.252. The initial value from the table is 30.

8. a. What is the base of the exponential model? Is the base a growth or decayfactor? How do you know?

The base, b, is 1.0394. It is greater than 1, so it is a growth factor.

b. What is the annual growth rate?

0.0394, or 3.94%

9. a. Use the exponential model to determine the number of college graduatesin 2010 1t � 1102.

thousand graduates

b. Use the exponential model to write an equation to determine the year inwhich there will be 2 million college graduates. Remember that the num-ber of college graduates is measured in thousands.

c. Solve the equation in part a using a graphing approach. Use the intersectfeature of your graphing calculator; the screen containing the solutionshould appear as follows.

10. What is the doubling time for your exponential model? That is, approxi-mately how many years will it take for a given number of college graduatesto double?


N � 40.2511.0394 2t � 2000

N � 40.25 # 1.0394110 � 2824

N � 40.2511.0394 2t


DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 342

Decreasing Exponential Model

Students in U.S. public schools have had much greater access to computers inrecent years. The following table shows the number of students per computerin selected years.


YEAR 1983 1984 1985 1987 1989 1992 1995 1999

NUMBER OF STUDENTS 125 75 50 32 22 16 10 5.7PER COMPUTER

11. a. Use your graphing calculator to determine the regression equation of anexponential function that models the given data. Let your input, t, repre-sent the number of years since 1983.

b. Sketch a graph of the exponential model using your graphing calculator.

c. What is the base of the exponential model? Is the base a growth or decayfactor? How do you know?

The base of the model is 0.837. The base is less than 1, so it is a decayfactor.

d. What is the annual decay rate?

0.163, or 16.3%

e. Does the graph have a horizontal asymptote? What is the practical mean-ing of this asymptote in the context of the situation?

The horizontal asymptote is C � 0. The number of students percomputer is becoming smaller.

C � 82.910.837 2t


1. The total amount of money spent on health care in the United States is increas-ing at an alarming rate. The following table gives the total national health careexpenditures in billions of dollars in selected years from 1975 through 2003.


YEAR 1975 1980 1985 1990 1995 2000 2003

TOTAL SPENT (billions 129.8 245.8 426.5 695.6 990.2 1309.9 1679.9of dollars)

Source: National Center for Health Statistics.

Billions for Health

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 343

a. Would the data in the preceding table be better modeled by a linearmodel, or an exponential model, ? Explain.

The data is not linear. The output seems to be increasingexponentially.

b. Sketch a scatterplot of this data.

c. Does the graph reinforce your conclusion in part a? Explain.

Yes; the scatterplot looks like an exponential function.

d. Use your graphing calculator to determine the exponential regressionequation that best fits the health care data in the preceding table. Let yourinput, t, represent the number of years since 1975.

e. Using the regression equation from part d, determine the predicted totalhealth care expenditures for the year 1995.

, or $919,550,000,000

f. According to the exponential model, what is the growth factor for thetotal health care costs per year?

The growth factor, b, is 1.093.

g. What is the growth rate?

0.093, or 9.3%

h. According to the exponential model, in what year did the total heath carecosts first exceed $1 trillion?

1996

i. What is the doubling time for your exponential model?


1t � 21 2

C � 155.3 # 11.093 220 � 919.55

C � 155.3 # 1.093t

600800

1000120014001600

200400

1975 1980 1985 1990 1995 2000x

Year

Hea

lth

Car

e S

pen

din

g(b

illio

ns)

y

y � ab xy � mx � b,


DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 344

2. a. Consider the following data set for the variables x and y.

x 5 8 11 15 20

y 70.2 50.7 35.1 22.6 9.5

Plot these points on the following grid.

b. Use your graphing calculator to determine both a linear regression and anexponential regression model of the data. Record the equations for thesemodels here.

linear:

exponential:

c. Which model appears to fit the data better? Explain.

The graph of the exponential model is closer to the points on thescatterplot.

d. Use the better model to determine y when x � 13 and y when x � 25.

e. For the exponential model, what is the decay factor?

0.877

f. What does it mean that the decay factor is between 0 and 1?

As the input increases, the output will decrease.

g. What is the half-life for the exponential model?

approximately 5.28

y � 144.2110.877 225 � 5.40

y � 144.2110.877 213 � 26.18

y � 144.2110.877 2xy � �3.95x � 84.20

50

70

90

10

30

20 6 10 14 184 8 12 16 20x

y


DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 345

3. Use the graph of as a model, and summarize the properties of theexponential function where

a. What is the domain?

the set of all real numbers

b. What is the range?

the set of all positive real numbers

c. When is positive?

is positive for all values of x.

d. When is negative?

is never negative.

e. What is the vertical intercept of the graph of ?

10, a2y � abx

y � a # b x

y � ab x

y � a # b x

y � ab x

a 7 0.y � ab x,

y � 512 2 x346 Chapter 3 Exponential and Logarithmic Functions

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 346

1. Consider a linear function defined by and an expo-nential function defined by Explain how you can determinefrom the equation whether the function is increasing or decreasing.

The function g is increasing if f is increasing if g isdecreasing if f is decreasing if

2. Suppose you have an exponential function of the form whereand and By inspecting the graph of f, can you deter-

mine if or if ? Explain.

Yes, if the function is increasing. If the function is decreasing,

3. You are given a function defined by a table, and the input values are inincrements of 1. By looking at the table, can you determine whether or notthe function can be approximated by an exponential model? Explain.

Yes, if the ratio between consecutive output values is roughly constant,the function can be approximated by an exponential function.

4. Explain the difference between growth rate and growth factor.

If the growth rate is r, the growth factor is The growth rate is theincrease in output per unit increase in input. The growth factor is thepercentage common factor that produces the next output (per unitincrease of input).

5. An exponential function passes through the point 10, 2.62. What canyou conclude about the values of a and b?

; nothing can be concluded about the value of b without furtherinformation.

6. You have just received a substantial tax refund of P dollars. You decide toinvest the money in a CD for 2 years. You have narrowed your choices to twobanks. Bank A will give you 6.75% interest compounded quarterly. Bank Boffers you 6.50% compounded continuously. Where do you deposit yourmoney? Explain.

The growth factor for bank A is The growth

factor for bank B is Therefore, I would deposit in bankA because the growth factor is larger.

7. Explain why the base in an exponential function cannot equal 1.

If the function would be a constant function because forall values of n.

1n � 1b � 1,

e0.065 � 1.06716.

11 �0.0675

4 24 � 1.06923.

a � 2.6

y � ab x

1 � r.

0 6 b 6 1.b 7 1,

0 6 b 6 1b 7 1b � 1.b 7 0a 7 0

f 1x 2 � abx,

0 6 b 6 1.m 6 0.b 7 1.m 7 0.

f 1x 2 � abx.m � 0,g1x 2 � mx � b,

Cluster 1 What Have I Learned? 347

CLUSTER 1 What Have I Learned?

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 347

1. You are planning to purchase a new car and have your eye on a specificmodel. You know that new car prices are projected to increase at a rate of 4%per year for the next few years.

a. Write an equation that represents the projected cost, C, of your dream cart years in the future, given that it costs $17,000 today.

b. Identify the growth rate and the growth factor.

The growth rate is 0.04; the growth factor is 1.04.

c. Use your equation in part a to project the cost of your car 3 years fromnow.

d. Use your graphing calculator to approximate how long it will take foryour dream car to cost $30,000, if the price continues to increase at 4%per year.

14.5 yr.

2. Without using your graphing calculator, match the graph with its equation.

a. b.

i.

Graph i is function g.

x

y

h1x 2 � 1.511.47 2xg1x 2 � 2.510.47 2x

C � 17,00011.04 23 � 19,122.69

C � 17,00011.04 2t


CLUSTER 1 How Can I Practice?


ii.

Graph ii is function h.

x

y

3. Explain the reasons for your choices in Problem 2.

Graph i is function g because it is decreasing and the only growth fac-tor between 0 and 1 is 0.47 in function g.

Graph ii is function h because it is positive and increasing with thegrowth factor of 1.47.

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 348

c.

4. Complete the following tables representing exponential functions. Roundcalculations to two decimal places whenever necessary.

a.

Cluster 1 How Can I Practice? 349

x 0 1 2 3 4

y 2.00 5.10 13.01 33.18 84.61

b. x 0 1 2 3 4

y 3.50 2.10 1.26 0.76 0.46

x 0 1 2 3 4

y 6 216 7776 279,93616

5. Write the equation of the exponential function that represents the data ineach table in Problem 4.

a. b. c.

6. Without graphing, classify each of the following functions as increasing ordecreasing, and determine f 102. (Use your graphing calculator to verify.)

y �16 136 2xy � 3.510.6 2xy � 212.55 2x

a.; decreasing

b.; increasingf 10 2 � 0.6

f 1x 2 � 0.611.03 2xf 10 2 � 1.3

f 1x 2 � 1.310.75 2x

c.

; decreasing

7. a. Given the following table, do you believe that it can be approximatelymodeled by an exponential function?

Yes

f 10 2 � 3

f 1x 2 � 3a15bx

x 0 1 2 3 4 5 6

y 2 5 12.5 31.3 78.1 195.3 488.3

b. If you answered yes to part a, what is the constant ratio of successive out-put values?

The constant ratio is 2.5, or

c. Determine an exponential equation that models this data.

8. Your starting salary for a new job is $22,000 per year. You are offered twooptions for salary increases:

Plan 1: an annual increase of $1000 per year or

Plan 2: an annual percentage increase of 4% of your salary

Your salary is a function of the number of years of employment at your job.

y � 212.5 2x

52.

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 349

a. Write an equation to determine the salary, S, after x years on the job usingplan 1; using plan 2.

Plan 1:

Plan 2:

b. Complete the following table using the equations from part a.

S � 22,00011.04 2xS � 22,000 � 1000x


x 0 1 3 5 10 15

S, PLAN 1 $22,000 $23,000 $25,000 $27,000 $32,000 $37,000

S, PLAN 2 $22,000 $22,800 $24,747 $26,766 $32,565 $39,621

c. Which plan would you choose? Explain.

It depends. If I plan to be with the company less than 10 years, Iwould take plan 1, because it takes plan 2 about 9 years to catch up.If I expect to be with the company for a long time, say 20 years, Iwould choose plan 2, because by then I would be better off by morethan $6000 per year.

9. The number of victims of a flu epidemic is increasing at a continuous rate of7.5% per week.

a. If 2000 people are currently infected, write an exponential model of theform where

N is the number of victims in thousands,

N0 is the initial number infected in thousands,

r is the weekly percent rate expressed as a decimal, and

t is the number of weeks.

b. Use the exponential model to predict the number of people infected after8 weeks.

thousand

c. Sketch the graph of the flu function using your graphing calculator.

d. Use a graphing approach to predict when the number of victims of the fluwill triple.

14.6482, or 15 wk.

N � f 18 2 � 2e0.075182 � 3.6442

N � f 1t 2 � 2e0.075t

N � f 1t 2 � N0ert,

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 350

10. a. Complete the following tables.


x �3 �2 �1 0 1 2 3

1 4 16 6414

116

164h(x) � 4x

x �3 �2 �1 0 1 2 3

64 16 4 1 164

116

14g(x) � (14)x

b. Sketch graphs of h and g on the following grid.

c. Use the tables and graphs in part a and b to complete this table.

1 2

–2

8

4

12

16

20

–1–2–3 3

h(x) = 4xg(x) =x1–

4

y

x


4 growth none 10, 12 x-axis increasing

decay none 10, 12 x-axis decreasing14g1x 2 � 1142x

h1x 2 � 4x

11. You are a college freshman and have a credit card. You immediately pur-chase a stereo system for $415. Your credit limit is $500. Let’s assume thatyou make no payments and purchase nothing more and there are no otherfees. The monthly interest rate is 1.18%.

a. What is your initial credit-card balance?

$415

b. What is the growth rate of your credit-card balance?

0.0118, or 1.18% per month

c. What is the monthly growth factor of your credit-card balance?

1.0118

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 351

d. Write an exponential function to determine how much you will owe1represented by f 1x2 2 after x months with no more purchases or payments.

e. Use your graphing calculator to graph this function. What is the verticalintercept?

10, 4152

f. What is the practical meaning of this intercept in this situation?

This represents the initial balance on the card.

g. How much will you owe after 10 months? Use the table feature on yourgraphing calculator to determine the solution.

$466.65

h. When you reach your credit limit of $500, the bank will expect apayment. How long do you have before you will have to start paying themoney back? Use the trace feature on your grapher to approximate thesolution.

With no payments, I exceed my credit limit during the sixteenthmonth.

12. You are working part-time for a computer company while going to college.The following table shows the hourly wage, w 1t2, in dollars, that you earn asa function of time, t. Time is measured in years since the beginning of 2000when you started working.

f 1x 2 � 41511.0118 2x


TIME, t, YEARS, SINCE 2000 0 1 2 3 4 5

HOURLY WAGE, w11t22 , ($) 12.50 12.75 13.01 13.27 13.53 13.81

a. Calculate the ratios of the outputs to determine if the data in the table isexponential. Round each ratio to the nearest hundredth.

The ratios are all 1.02, so the data is exponential.


1.02

c. Write an exponential equation that models the data in the table.

d. What percent raise did you receive each year?

2%

e. If you continue to work for this company, what can you expect yourhourly wage to be in 2010?

$15.24

w 1t 2 � 12.5011.02 2x

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 352

f. For approximately how many years will you have to work for the com-pany in order for your hourly wage to double? (Assume you will receivethe same percentage increase each year.)

about 35 yr.

13. You deposited $10,000 in an account that pays 12% annual interest com-pounded monthly.

a. Write an equation to determine the amount, A, you will have in t years.

b. How much will you have in 5 years?

$18,166.97

c. Use your graphing calculator to determine in how many years yourinvestment will double.

approximately 6 yr.

d. Write an equation to determine the amount, A, you will have in t years ifthe interest is compounded continuously.

e. Use the equation in part d to determine how much you will have in5 years. Compare your answer to your answer in part b.

$18,221.19; $54.22 more than in part b

14. The number of farms in the United States has declined from 1940 to 2000,as the data in the following table shows. The data is estimated from NationalAgricultural Statistics Service, U.S. Department of Agriculture.

A � 10,000e0.12t

A � 10,00011.01 212t

A � 10,00011 �0.1212 212t


YEAR 1940 1950 1960 1970 1980 1990 2000

NUMBER OF FARMS, 6.2 5.8 4 3 2.5 2.2 2IN MILLIONS

a. Make a scatterplot of this data.

2

4

6

8

10

1940 1960 1980 20001950 1970 1990

Year

Nu

mb

er o

f Fa

rms

(in

mill

ion

s)

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 353

b. Does the scatterplot show that the data would be better modeled by a lin-ear model or by an exponential model? Explain.

An exponential decay model would better model the data. The datais decreasing, but not at a constant rate.

c. Use your graphing calculator to determine the exponential regressionequation that best fits the U.S. farm data. Let x represent the number ofyears since 1940.

d. Use the regression equation to predict the total number of farms in theU.S. in 2010.

million farms

e. According to your exponential model, what is the decay factor for thetotal number of farms in the United States?

0.9795

f. What is the decay rate?

, or 2.05%

g. Explain the meaning of the decay rate found in part f in this situation.

The number of farms is decreasing at a rate of 2.05% per year.

h. Use your graphing calculator to determine the halving time for yourexponential model.

34 yr.

r � 1 � 0.9795 � 0.02050.9795 � 1 � r;

6.2410.9795 270 � 1.46

y � 6.2410.9795 2x


DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 354

ACTIVITY 3.8

The Diameter ofSpheres

OBJECTIVES

1. Define logarithm.

2. Write an exponentialstatement in logarithmicform.

3. Write a logarithmic state-ment in exponential form.

4. Determine log and lnvalues using a calculator.

CLUSTER 2 Logarithmic Functions

Spheres are all around you (pardon the pun). You play sports with spheres likebaseballs, basketballs, and golf balls. You live on a sphere. Earth is a big ballin space, as are the other planets, the Sun, and the Moon. All spheres have proper-ties in common. For example, the formula for the volume, V, of any sphere is

and the formula for the surface area, S, of any sphere is

where r represents the radius of the sphere.

However, not all spheres are the same size. The following table gives the diame-ter, d, of some spheres you all know. Recall that the diameter, d, of a sphere istwice the radius, r.

r

S � 4pr 2V �

43pr

3,

SPHERE DIAMETER, d, IN METERS

Golf ball 0.043

Baseball 0.075

Basketball 0.239

Moon 3,476,000

Earth 12,756,000

Jupiter 142,984,000

If you want to determine either the volume or surface area of any of the spheres inthe preceding table, the diameter of the given sphere would be the input value andwould be referenced on the horizontal axis. But how would you scale this axis?

1. a. Plot the values in the first three rows of the table. Scale the axis starting at0 and incrementing by 0.02 meter.

b. Can you plot the values in the last three rows of the table on the sameaxis? Explain.

No, the scale is so small I run out of paper.

0 .02 .04 .06 .08 .10 .12 .14 .16 .18 .20 .22 .24

Activity 3.8 The Diameter of Spheres 355

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 355

2. a. Plot the values in the last three rows of the table on a different axis. Scalethe axis starting at 0 and incrementing by 10,000,000 meters.


20,000,000 40,000,000 60,000,000 80,000,000 100,000,000 120,000,000 140,000,000

b. Can you plot the values in the first three rows of the table on the axis inpart a? Explain.

Not very easily. The scale is too big.

Logarithmic Scale

3. There is a way to scale the axis so that you can plot all the values in the tableon the same axis.

a. Starting with the leftmost tick mark, give the first tick mark a value ofmeter. Give the next tick mark a value of meter. Continue in

this way by giving each consecutive tick mark a value that is one power of10 greater than the preceding tick mark.

b. Complete the following table by writing all of the diameters from the pre-ceding table in scientific notation.

100

Golf ball

Baseball

Basketball

MoonEarth

Jupiter

101 10210–2 10–1 103 104 105 106 107 108 109

10�110�2

SPHERE DIAMETER, d, IN METERS d, IN SCIENTIFICNOTATION

Golf ball 0.043

Baseball 0.075

Basketball 0.239

Moon 3,476,000

Earth 12,756,000

Jupiter 142,984,000 1.4298 � 108

1.2756 � 107

3.476 � 106

2.39 � 10�1

7.5 � 10�2

4.3 � 10�2

c. To plot the diameter of a golf ball, notice that 0.043 meter is between

meter and meter. Now using the axis in part a,

plot 0.043 meter between the tick mark labeled and meter,

closer to the tick mark labeled meter.

d. To plot the diameter of Earth, notice that 12,756,000 meters is between

meters and meters. Now plot

12,756,000 meters between the tick marks labeled and meters,

closer to the tick mark labeled .107108107

108 � 100,000,000107 � 10,000,000

10�210�110�2

10�1 � 0.110�2 � 0.01

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 356

e. Plot the remaining data in the same way by first determining betweenwhich two powers of 10 the number lies.


The scale you used to plot the diameter values is a logarithmic or log scale. Thetick marks on a logarithmic scale are usually labeled with just the exponent of thepowers of 10.

4. a. Rewrite the axis from Problem 3a by labeling the tick marks with just theexponents of the powers of 10.

b. The axis looks like a standard axis with tick marks labeled �2, �1, 0, 1,etc. However, it is quite different. Describe the difference between this logscale and a standard axis labeled in the same way. Focus on the valuesbetween consecutive tick marks.

Each unit distance increases the power of 10 by 1 rather than anaddend of 1.

–2 –1 0 1 2 3 4 5 6 7 8 9 10

DEFINITION

The exponents used to label the tick marks of the preceding axis are logarithmsor simply logs. Since these are exponents of powers of 10, the exponents are logsbase 10, known as common logarithms or common logs.

Example 1

a. The common logarithm of is the exponent to which 10 must be raised toobtain a result of . Therefore, the common log of is 3.

b. The common log of is �2.

c. The common log of is 2.

5. Determine the common log of each of the following.

100 � 102

10�2

103103103

a.�1

b.4

10410�1 c. 1000

3

d. 100,000

5

e. 0.0001

�4

Logarithmic Notation

Remember that a logarithm is an exponent. The common log of x is anexponent, y, to which the base, 10, must be raised to get result x. That is, in theequation 10y � x, y is the logarithm. Using log notation log10 x � y. Therefore,log10 10,000 � log10 104 � 4.

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 357

Example 2


x, THE NUMBER y, THE EXPONENT TO LOG NOTATION LOG10 x � yWHICH THE BASE, 10,

MUST BE RAISED TO GET x

3

�2

100 2 log10 100 � 2

log10 10�2 � �210�2

log10 103 � 3103

When using logs base 10, the notation is shortened by dropping the 10.Therefore,

;

6. Determine each of the following:

log10 100 � log 100 � 2log10 103 � log 103 � 3

log10

a.�1

b.4

log 104log 10�1 c.3

log 1000

d.5

e.�4

log 0.0001log 100,000

Bases for Logarithms

The logarithmic scale for the diameter of spheres situation was labeled with theexponents of powers of 10. Using 10 as the base for logarithms is common sincethe number 10 is the base of our number system. However, other numbers couldbe used as the base for logs. For example, you could use exponents of powers of 5or exponents of powers of 2.

Example 3

Base-5 logarithms: The log base 5 of a number, x, is the exponent to which the base, 5,must be raised to obtain x. For example,

a.

b.

c.

Base-2 logarithms: The log base 2 of a number, x, is the exponent to which 2 must beraised to obtain x. For example,

a.

b.

c. log2 18 � log2 2�3 � �3

log2 16 � log2 24 � 4

log2 25 � 5

log5 125 � log5 5�2 � �2

log5 125 � log5 53 � 3

log5 54 � 4

In general, a statement in logarithmic form is where b is the base ofthe logarithm, x is a power of b, and y is the exponent. The base b for a logarithmcan be any positive number except 1.

logb x � y,

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 358

7. Determine each of the following.


b.�4

log2 116a.

3

log4 64

d.�3

log3 127c.

2

log3 9

The examples and problems so far in this activity demonstrate the followingproperty of logarithms.

Property of Logarithms

For any base b 1b 0, b � 12, .logb bn � n

8. Determine each of the following:

a.0

b.0

c.0

log12 1log5 1log 1

f.1

log1>2 1122d.1

log 10 e.1

log5 5

9. a. Referring to Problem 8a–c, write a general rule for .

b. Referring to Problem 8d–f, write a general rule for .

logb b � 1

logb b

logb 1 � 0

logb 1

Property of Logarithms

In general, and where b 7 0, b � 1logb b � 1,logb 1 � 0

Natural Logarithms

Because the base of a log can be any positive number except 1, the base can be thenumber e. Many applications involve the use of log base e. Log base e is calledthe natural log and has the following special notation:

is written as read simply as el-n-x.

Example 4

a. b. ln 1e4 � ln e�4 � �4ln e2 � loge e

2 � 2

ln x,loge x

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 359

10. Evaluate the following.


b.�3

ln 1 1e32a.7

ln e7 c.0

ln 1

d.1

e.12

ln 2eln e

Logarithmic and Exponential Forms

Because logarithms are exponents, logarithmic statements can be written asexponential statements, and exponential statements can be written as logarith-mic statements.

For example, in the statement , the base is 5, the exponent (loga-

rithm) is 3, and the result is 125. This relationship can also be written as the

equation .53 � 125

3 � log5 125

In general, the logarithmic equation is equivalent to the exponentialequation .b

y � xy � logb x

Example 5 Rewrite the exponential equation as an equivalent loga-

rithmic equation.

SOLUTION

In the equation , the base is e, the result is x, and the exponent (logarithm) is0.5. Therefore, the equivalent logarithmic equation is , or

11. Rewrite each exponential equation as a logarithmic equation and each logequation as an exponential equation.

0.5 � ln x.0.5 � loge xe0.5 � x

e0.5 � x

c.

2�4 �116

log2 116 � �4a.

23 � 8

3 � log2 8 b.

e3 � e3

ln e3 � 3

d. e. f.

log3 19 � �2

3�2 �19

ln e � 1

e1 � e

log6 216 � 3

63 � 216

Logarithms and the Calculator

The numbers whose logarithms you have been working with have been exactpowers of the base. However, in many situations, you have to evaluate a logarithmwhere the number is not an exact power of the base. For example, what is log 20or ln 15? Fortunately, the common log (base 10) and the natural log (base e) areon your calculator.

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 360

1. The notation for logarithms is , where b is the base of the log,x is the resulting power of b, and y is the exponent. The base b can be anypositive number except 1; x can be any positive number. The range of yvalues includes all real numbers.

2. The notation for common logarithm or base-10 logarithms is

3. The notation for the natural logarithm or base e logarithm is

4. The logarithmic equation is equivalent to the exponentialequation b

y � x.y � logb x

loge x � ln x.

log10 x � log x.

logb x � ySUMMARYACTIVITY 3.8

12. Use your calculator to evaluate the following.

a.1.3010

log 20


b.2.7081

ln 15

c.�.6931

ln 12 d.�1.6990

log 0.02

e. Use your calculator to check your answers to Problems 6 and 10.

They check.

13. a. Use your calculator to complete the following table. Confirm the placementof the diameter values on the log-scaled axis.

Golf ball 0.043 �1.3665

Baseball 0.075 �1.1249

Basketball 0.239 �0.6216

Moon 3,476,000 6.5411

Earth 12,756,000 7.1057

Jupiter 142,984,000 8.15531.4298 � 108

1.2756 � 107

3.476 � 106

2.39 � 10�1

7.5 � 10�2

4.3 � 10�2

SPHERE DIAMETER, d, IN METERS d, IN SCIENTIFIC log(d)NOTATION

b. Plot the values from the log column in the preceding table on thefollowing axis.

c. Compare the preceding plot with the plot in Problem 3a and comment.

Answers will vary. The locations of the points are the same. The scaleis units of 1 rather than powers of 10. However each successive tickmark in the graph above represents the corresponding power of 10.

–1.4 –1.1 –0.6 6.5 7.1 8.1

0 1–1–2 2 3 4 5 6 7 8 9

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 361

5. If and ,

a.b.c. logb bn � n

logb b � 1

logb 1 � 0

b � 1b 7 0




1. Use the definition of logarithm to determine the exact value of each of thefollowing.

a.5

b.3

c.�1

log 0.1log3 27log2 32

d.�6

log2 1 1642g.

Hint:12

27 � 7 1>221log7 27 h.

12

log100 10 i.

0

log 1

j.0

log2 1

e.0

log5 1 f.2

log1>2 1142

k.5

ln e5 l.�2

ln 1 1e22 m.0

ln 1

2. Evaluate each common logarithm without the use of a calculator.

a. �3log a 11000

b � b. �2log a 1100b �

c. �1log a 110b � d. 0log 1 �

e. 1log 10 � f. 2log 100 �

g. 3log 1000 � h. ln 12

210

3. Rewrite the following equations in logarithmic form.

a.

log3 9 � 2

32 � 9 b. (Hint: First rewrite in exponential form.)

log121 11 �12

21212121 � 11

c.log4 27 � t

4t � 27 d.logb 19 � 3

b3 � 19

4. Rewrite the following equations in exponential form.

a.34 � 81

log3 81 � 4 b.

1001>2 � 10

12 � log100 10

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 362

c.912 � N

log9 N � 12


d.7y � x

y � log7 x

e.

e1>2 � 2e

ln 2e �12 f.

e�2 �1e2

ln 1 1e22 � �2

5. Estimate between what two integers the solutions for the following equa-tions fall. Then solve each equation exactly by changing it to log form. Useyour calculator to approximate your answer to three decimal places.

a. b. c.

x � �5.347

x � log 0.0000045

�6 6 x 6 �5

10 x � 0.0000045

x � 2.771

x � log 590

2 6 x 6 3

10 x � 590

x � 0.512

x � log 3.25

0 6 x 6 1

10 x � 3.25

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 363

On a recent visit to Boston, you notice that people seem rushed as they moveabout the city. Upon returning to college, you mention this observation to yourpsychology instructor. The instructor refers you to a psychology study that inves-tigates the relationship between the average walking speed of pedestrians and thepopulation of the city. The study cites statistics presented graphically as follows.

1. a. Does the data appear to be linear? Explain.

No, the rate of change is not constant.

b. Does the data appear to be exponential? Explain.

No, it is growing but slower rather than faster as x increases.

This data is actually logarithmic. Situations that can be modeled by logarithmicfunctions will be the focus of this and the following activity.

Introduction to the Logarithmic Function

The logarithmic function base b is defined by where

b represents the base of the log ,

x is the input and represents a power of the base b (x is also called theargument), and y is the output and is the exponent needed on the base b toobtain x.

2. a. Evaluate using your calculator. What do you observe?Does it seem reasonable? Explain.

I get an error that is not a real answer. Yes, it seems reasonablebecause 10 raised to any power will never be negative.

b. Is it possible to determine log 102?No, because 10 raised to any power will never be 0.

c. What is the domain for the function defined by ?

x 7 0

y � log x

log10 1�100 2

1b 7 0, b � 1 2y � logb x,

Ave

rage

Wal

king

Spe

ed, S

(ft

./sec

.)

Population, P (in thousands)

00 200 400 600 800 1000

(50, 3.70)

(100, 4.30)

(200, 4.90) (500, 5.70)(1000, 6.30)

(2000, 6.90)

1200 1400 1600 180 2000

1

2

3

4

5

6

7

8

On the Move


ACTIVITY 3.9

Walking Speedof Pedestrians

OBJECTIVES

1. Determine the inverse ofthe exponential function.

2. Identify the properties ofthe graph of a logarithmicfunction.

3. Graph the naturallogarithmic function.

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 364

d. What is the range? Remember, the output y is an exponent.

The range is all real numbers, since 10y is defined for any y.

3. The exponential function defined by has a special relationship withthe corresponding logarithmic function defined by

a. Complete the following tables for and g 1x 2 � log x.f 1x 2 � 10 x

g1x 2 � log10 x � log x.f 1x 2 � 10

x

Activity 3.9 Walking Speed of Pedestrians 365

�2 0.01

�1 0.1

0 1

1 10

2 100

x f 11x22 � 10x

0.01 �2

0.1 �1

1 0

10 1

100 2

x g 11x22 � log x

b. Compare the input and output values for functions f and g.

The inputs and outputs are interchanged.

c. Sketch the graphs of and using your graphingcalculator. Use the window Xmin � �4, Xmax � 4, Ymin � �3, andYmax � 3. Your screen should appear as follows.

d. Graph on the same coordinate axes as functions f and g. Describe ina sentence or two the symmetry you observe in the graphs of f and g.

The graphs are reflected over the line

Recall the concept of an inverse function from Chapter 2. The inverse functioninterchanges the domain and range of the original function. Also, the graph of aninverse function is the reflection of the original function about the line y � x. There-fore, it appears from the results in Problem 3 that and areinverse functions.

You can determine the equation of the inverse function by interchanging the input(x-values) and the output (y-values) in the given equation for the function andsolving the new equation for y.

g1x 2 � log xf 1x 2 � 10 x

y � x.

y � x

Y2 � log10 xY1 � 10 x

Example 1 Determine the equation of the inverse of the function defined by

SOLUTION

Step 1. Interchange the x and y variables.

Step 2. Solve the resulting equation for y bywriting the statement in logarithmic notation.

y � log5 x

x � 5y

y � 5x.

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 365

4. Use the algebraic approach demonstrated in Example 1 to verify that is the inverse of

Start with Interchange x and y.

Write in exponential form.

Problems 2, 3, and 4 illustrate the following properties of the common logarithmicfunction.

y � 10 x

x � log y.

y � log x.

y � 10 x.

y � log x


Properties of the Common Logarithmic Function f 11x22� log x

1. The domain of f is the set of all positive real numbers

2. The range of f is all real numbers.

3. f is the inverse of the function defined by g 1x 2 � 10 x.

1x 7 0 2.

The Graph of the Natural Logarithmic Function

5. a. Using your calculator, complete the following table. Round your answersto three decimal places.

x 0.1 0.5 1 5 10 20 50

y � ln x �2.303 �0.693 0 1.609 2.303 2.996 3.912

b. Sketch a graph of

5 10

–1

–2

1

2

3

15 20 25

y = lnx

y

x

y � ln x.

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 366

c. Verify your graph in part b using your graphing calculator. Using thewindow Xmin � �1, Xmax � 4, Ymin � �2.5, and Ymax � 2.5, yourscreen should appear as follows.

d. What are the domain and range of the function defined by ?

The domain is The range is all real numbers.

e. Does the graph of have a horizontal asymptote? Explain.

No; as x increases in value, the output continues to increase slowly.The output values eventually get infinitely large.

f. Does the graph of have a vertical intercept?

No, the log functions are undefined at x � 0.

g. Complete the following table using your calculator. Round your answersto the nearest tenth.

y � ln x

y � ln x

x 7 0.

y � ln x


x 1 0.5 0.25 0.1 0.01 0.001

y � ln x 0 �0.7 �1.4 �2.3 �4.6 �6.9

h. As the input values take on values closer and closer to 0, what happens tothe corresponding output values?

The output values become very large negative numbers.

DEFINITION

A vertical asymptote is a vertical line, x � a, that the graph of a functionbecomes very close to but never touches. As the input values get closer and closerto x � a, the output values get larger and larger in magnitude. That is, the outputvalues become very large positive or very large negative values.

Example 2 The vertical asymptote of the graphs of y � log x and y � ln x isthe vertical line x � 0.

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 367

6. a. Graph and y � x on the same set of coordinate axes usingthe window Xmin � �7.5, Xmax � 7.5, Ymin � �5, and Ymax � 5.Describe the symmetry that you observe.

The graphs are symmetric in the line y � x.

b. Use an algebraic approach to determine the inverse of the exponentialfunction defined by

Start with y � . Interchange x and y.

x � . Write in logarithmic form.

y � ln x

e y

e x

y � e x.

y � ln x,y � e x,


1. Properties of the log function defined by

a. The domain of f is

b. The range of f is all real numbers.

c. f is the inverse of the function defined by

2. The graph of a logarithmic function defined by where resembles the following graph, and the function

a. is increasing for all

b. has an x-intercept of 11, 02c. has no y-intercept

d. has a vertical asymptote of the y-axis

3. The natural logarithmic function is defined by

y � ln x � loge x.

y

x2

3

1

2

4 6 8

–1

–2

x � 0,

x 7 0

b 7 1,y � logb x,

g1x 2 � 10 x.

x 7 0.

y � log x.SUMMARYACTIVITY 3.9

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 368

4. The graph of the natural logarithmic function

a. is increasing for all

b. has an x-intercept of 11, 02c. has a vertical asymptote of the y-axis

5. You can determine the equation of the inverse of the function by interchangingthe input (x-values) and the output (y-values) in the given equation for thefunction and solving the new equation for y.

x � 0,

x 7 0

y � ln x




1. Using the graph of as a check, summarize the following properties ofthe common logarithmic function.




c. For what values of x is log x positive?

d. For what values of x is log x negative?

e. For what values of x does log ?

f. For what values of x does ?

2. a. Complete the following table using your calculator. Round your answers tothe nearest tenth.

x � 10

log x � 1

x � 1

x � 0

0 6 x 6 1

x 7 1

x 7 0

y � log x

x 0.001 0.01 0.1 0.25 0.5 1

y � log x �3 �2 �1 �0.6 �0.3 0

b. As the positive input values take on values closer and closer to 0, whathappens to the corresponding output values?

The output values become very large negative numbers (large inabsolute value).

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 369

c. Determine the vertical asymptote of the graph of

the y-axis

3. The exponential function defined by has an inverse. Determine theequation of the inverse function. Write your answer in logarithmic form.

Start with Interchange x and y.

Rewrite in log form.

4. Using the graph of as a check, summarize the following propertiesof the natural logarithmic function.




c. For what values of x is positive?

d. For what values of x is negative?

e. For what values of x does ?

f. For what values of x does ?

5. The life expectancy for a piece of equipment is the number, n, of years forthe equipment to depreciate to a known salvage value, V. The life ex-pectancy, n, is given by the formula

where C is the initial cost of the piece of equipment and r is the annual rateof depreciation expressed as a decimal. If a computer costs $34,000 and hasa salvage value of $1000, what is the life expectancy if the annual rate ofdepreciation is 40%?

n �log 11000 2 � log 134,000 2

log 11 � 0.40 2 � 6.9 yr.

n �log V � log C

log 11 � r 2 ,

x � e

ln x � 1

x � 1

ln x � 0

0 6 x 6 1

ln x

x 7 1

ln x

x 7 0

y � ln x

y � log2 x

x � 2 y.

y � 2 x.

y � 2x

x � 0,

y � log x.


DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 370

In Activity 3.9 you looked at a psychology study that investigated the relationshipbetween the average walking speed of pedestrians and the population of the city.Graphically the data was presented as follows.

1. a. Does the data appear to be logarithmic? Explain.

Yes, it grows rather quickly at first, but for larger values of input thegrowth slows a lot.

b. Use the data in the graph to complete the following table.

Ave

rage

Wal

king

Spe

ed, S

(ft

./sec

.)

Population, P (in thousands)

00 200 400 600 800 1000

(50, 3.70)

(100, 4.30)

(200, 4.90) (500, 5.70)(1000, 6.30)

(2000, 6.90)

1200 1400 1600 180 2000

1

2

3

4

5

6

7

8

On the Move

Activity 3.10 Walking Speed of Pedestrians, continued 371

ACTIVITY 3.10

Walking Speedof Pedestrians,continued

OBJECTIVES

1. Compare the average rateof change of increasinglogarithmic, linear, andexponential functions.

2. Determine the regressionequation of a naturallogarithmic function thatbest fits a set of data.

POPULATION, P (in thousands) 50 100 200 500 1000 2000

AVERAGE WALKING SPEED, S 3.70 4.30 4.90 5.70 6.30 6.90

The natural logarithmic function can be used to model a variety of scientific and nat-ural phenomena. The natural logarithmic function is so prevalent that on mostgraphing calculators it has its own built-in regression finder.

2. a. Use your graphing calculator and the table in Problem 1b to produce ascatterplot of the average walking speed data.

b. Use the regression feature of your calculator to produce a naturallogarithmic curve that approximates the data in the table. Use option9 from the STAT CALC menu.

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 371

The LnReg option will generate a regression equation of the formRound a and b to the nearest thousandth, and record the

function below.

c. Enter the function from part b into your graphing calculator. Verifyvisually that this function is a good model for your data.

d. What is the practical domain of this function?

The domain is all real numbers greater than 0 and less than 9000.(Answers may vary.)

e. Use the function from part b to predict the average walking speed inBoston, population 589,121. (Note: P is in thousands (589.121 thousands).)

ft./sec.

f. Use the model to predict the average walking speed in New York City,population 8,008,278.

ft./sec.

3. a. If the average walking speed in a certain city is 5.2 feet per second, writean equation that can be used to estimate the population P of the city.

b. Solve the equation using a graphical approach.

The population is approximately 282,000.

5.2 = 0.303 + 0.868 ln P

100 200

2

4

6

8

300

(281.94, 5.2)

400 500

y

x

5.2 � 0.303 � 0.868 ln P

S � 0.303 � 0.868 ln 18008.278 2 � 8.10

S � 0.303 � 0.868 ln 1589.121 2 � 5.84

S � 0.303 � 0.868 ln P

y � a � b ln x.


DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 372

Comparing the Average Rate of Change of Logarithmic,Linear, and Exponential Functions

4. a. Complete the following table using the function defined byS � 0.303 � 0.868 ln P.


P, POPULATION (thousands) 10 20 150 250

S, AVERAGE WALKING SPEED (ft./sec.) 2.30 2.90 4.65 5.09

b. Determine the average rate of change of S as the population increasesfrom

i. 10 to 20 thousand

ft./sec./thousand

ii. 20 to 150 thousand

ft./sec./thousand

iii. 150 to 250 thousand

ft./sec./thousand

c. What can you say in general about the average rate of change in thewalking speed as the population increases?

The average rate of change in the walking speed decreases as thepopulation increases.

You should have discovered that the average rate of change in this situation isalways positive. This means that the walking speed increases as the populationincreases. Nevertheless, in general, the increase gets smaller as the populationincreases. This is characteristic of logarithmic functions.

5.09 � 4.65250 � 150 �

0.44100 � 0.0044

4.65 � 2.90150 � 20 �

1.75130 � 0.013

2.90 � 2.3010 � 0.06

As the input of a logarithmic function with increases, the output increasesat a slower rate (the graph becomes less steep).

b 7 1

5. Complete the following statements by describing the rate at which the outputvalues change.

a. For an increasing linear function, as the input variable increases, the output

b. For an increasing exponential function, as the input increases, the output

c. For an increasing logarithmic function, as the input increases, the outputincreases at a decreasing rate.

increases at an increasing rate.

increases at a constant rate.

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 373

6. Consider the graphs of


i. ii. iii. g1x 2 � ln xh1x 2 � xf 1x 2 � ex

using the window Xmin � �7.5, Xmax � 7.5, Ymin � �5, and Ymax � 5.

a. Which of the functions are increasing?

All three are increasing.

b. Which of the functions are decreasing?

None are decreasing.

c. As the input values get larger, which of the functions grows fastest?

d. As the input values get larger, which of the functions grows most slowly?

e. Do any of these functions have a horizontal asymptote?

has the horizontal asymptote

f. Do any of these functions have a vertical asymptote?

has the vertical asymptote

g. Compare the domains of these functions.

The domain of and is all real numbers. The domainof is

h. Compare the ranges of these functions.

The range of and is all real numbers. The range ofis

Problem 6 illustrates some of the relationships between where where and where

Application

7. You are working on the development of an “elastic” ball for the IBF ToyCompany. The question you are investigating is, “If the ball is launchedstraight up, how far has it traveled vertically when it hits the ground for the10th time?”

m 7 0.y � mx � b,b 7 1,g1x 2 � logb x,b 7 1,f 1x 2 � bx,

y 7 0.f 1x 2 � exh1x 2 � xg1x 2 � ln x

x 7 0.g1x 2 � ln xh1x 2 � xf 1x 2 � ex

x � 0.g1x 2 � ln x

y � 0.f 1x 2 � e x

g1x 2 � ln x

f 1x 2 � e x

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 374

Your launcher will project the ball 10 feet into the air. This means it willtravel 20 feet (10 feet up and 10 down) before it hits the ground the first time.Assuming that the ball returns to 50% of its previous height, it will rebound5 feet and travel 10 feet before it hits the ground again. The following tablesummarizes this situation.

10 ft

5 ft

1stNot to scale.

2nd 3rd 4th5th

2.5 ft

1.25 ft.0.625 ft.

0.3125 ft.0.15625 ft.0.078125 ft.0.0390625 ft.0.01953125 ft.

6th7th8th9th

10th


N, times the ball hits the ground 1 2 3 4 5 6

Distance traveled since last time (ft.) 20 10 5 2.5 1.25 0.625

T, Total distance traveled (ft.) 20 30 35 37.5 38.75 39.375

a. Using the window Xmin � 0, Xmax � 7, Ymin � 0, and Ymax � 45, aplot of N versus T should resemble the following.

b. Do the table and scatterplot indicate the data is linear, exponential, orlogarithmic?

The rate of change appears to be slowing, so the data should beapproximately logarithmic.

c. Use your graphing calculator to produce linear, exponential, and naturallog regression equations for the given data.

linear:

exponential:

natural log: T � 21.373 � 11.002 ln N

T � 21.37211.128 2NT � 3.589N � 20.875

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 375

increasing fastest horizontal all real numbersasymptote

increasing slowest vertical all real numbersasymptote

increasing constant none all real numbers all real numbersm 7 0y � mx � b,

b 7 1x 7 0g1x 2 � logb x,

b 7 1y 7 0f 1x 2 � bx,

d. Graph each equation and visually determine which of the regressionmodels best fits the data.

The natural log fits the best.

e. Use the equation of best fit to predict the total distance traveled by theball when it hits the ground for the 10th time.

ft.T110 2 � 21.373 � 11.002 ln 110 2 � 46.706


1. As the input of a logarithmic function increases, the output increases at aslower rate (the graph becomes less steep).

2. The relationships among the graphs where where and where are identified in the followingtable.

m 7 0,y � mx � b,b 7 1,g1x 2 � logb x,b 7 1,f 1x 2 � bx,

SUMMARYACTIVITY 3.10

GRAPHS INCREASING OR GROWTH RATE HORIZONTAL OR DOMAIN RANGEDECREASING VERTICAL

ASYMPTOTE


1. Chlamydia trachomatis infections are the most commonly reported notifiabledisease in the United States. These are among the most prevalent of all sexuallytransmitted diseases. The following data from the Centers for Disease Controland Prevention indicates the reported rates, R, in rates per 100,000 people from1985 to 2002. Let t represent the number of years since 1985.


YEARS, t, SINCE 1985 1 3 5 9 13 15 17

REPORTED RATES: U.S. 40 90 160 200 250 260 288(rate per 100,000 population), R

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 376

a. Plot the points on the following grid.

b. Does the scatterplot indicate that the data is logarithmic? Explain.

It could very well be logarithmic. It increases more slowly as the inputincreases.

c. Determine the natural log regression equation. Record the regressionequation below, and add a sketch of the regression curve to the scatterplotin part a.

d. Is the graph a good fit of the data?

Yes, it is a very good fit.

e. Use your model to predict the reported rate of Chlamydia trachomatisinfections per 100,000 people in 2010.

2010 is 25 years after 1985, so evaluate R when

per 100,000 population

2. a. Consider a data set for the variables x and y.

R � 19.946 � 88.259 ln 25 � 304

t � 25.

R � 19.946 � 88.259 ln t

80

160

240

320

400

0 5 10 15 17Years since 1985

Rat

e P

er 1

00,0

00 P

eop

le


x 1 4 7 10 13

f 11x22 3.0 4.5 5.0 5.2 5.8

Plot these points on the following grid.

3456

12

4 8 12 16x

f(x)

DANFMC03_0321447808 pp3 1/6/07 11:30 AM Page 377

b. Does the scatterplot indicate the data is more likely linear, exponential, orlogarithmic? Explain.

Logarithmic. It is increasing, but at continually lesser rates.

c. Use your graphing calculator to determine a logarithmic regression modelthat represents this data.

d. Use your model to determine f 1112 and f 1202.,

3. The barometric pressure, P, in inches of mercury at a distance x miles fromthe eye of a moderate hurricane can be modeled by

a. Determine f 102. What is the practical meaning of the value in this situation?

in. This is the pressure in the eye of the storm.

b. Sketch a graph of this function.

c. Describe how air pressure changes as you move away from the eye of thehurricane.

As you move away from the hurricane’s eye, the pressure increasesquickly at first and then more slowly.

P(x) = 0.48ln(x + 1) + 27

x

P(x)

//

28

29

27

5 10 15 20 25Distance from Eye (miles)

Bar

om

etri

c P

ress

ure

(in

. of

mer

cury

)

f 10 2 � 27

P � f 1x 2 � 0.48 ln 1x � 1 2 � 27.

f 120 2 � 6.1f 111 2 � 5.48

y � 3 � 1.033 ln x


DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 378

4. The formula is used to approximate the minimum re-quired ventilation rate, R, as a function of the air space per child in a publicschool classroom. The rate R is measured in cubic feet per minute, and x ismeasured in cubic feet.

a. Sketch a graph of the rate function for

b. Determine the required ventilation rate if the air space per child is 300 cubic feet.

R � 80.4 �11 ln 13002 � 1766 cu. ft.

5. You have recently accepted a job working in the coroner’s office of a largecity. Because of the large numbers of homicides, it has been very difficultfor the coroners to complete all of their work. Your job is, in part, to assistthem in the paperwork. On one particular day, you are working on a case inwhich you are attempting to establish the time of death.

The coroner tells you that to establish the time of death, he uses the formula

where t is the number of hours the victim has been dead,

Tb represents the temperature of the body when discovered, and

Ts represents the temperature of his surroundings.

The coroner also tells you that the thermostat was set at 68°F in the apart-ment in which the body was found and that the victim’s body temperaturewas 78°F.

a. Using the preceding formula, determine the number of hours the victimhas been deceased. Use your calculator to approximate your answer toone decimal place.

4.5 hr.

b. If the body was discovered at 10:07 P.M., what do you estimate for thetime of death?

5:37 P.M.

t � 4 ln 98.6 � Ts

Tb � Ts,

R = 80.4 – 11ln(x)

x

R

//

15202530

510

100 500 900 1300 1700Air Space per Child (cu. ft.)

Rat

e o

f V

enti

lati

on

(cu

. ft.

/min

.)

100 � x � 1500.

R � 80.4 � 11 ln x


DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 379


ACTIVITY 3.11

The Elastic Ball

OBJECTIVES

1. Apply the log of aproduct property.

2. Apply the log of aquotient property.

3. Apply the log of a powerproperty.

4. Discover change of baseformula.

You are continuing your work on the development of the elastic ball. You are stillinvestigating the question, “If the ball is launched straight up, how far has it trav-eled vertically when it hits the ground for the 10th time?” However, your supervi-sor tells you that you cannot count the initial launch distance. You must calculateonly the rebound distance.

Using some physical properties, timers, and your calculator, you collect the fol-lowing data.

1stNot to scale.

2nd 3rd 4th5th6th7th8th9th

10th

N, NUMBER OF TIMES THE BALL 1 2 3 4 5 6HITS THE GROUND

T, TOTAL REBOUND 0 9.0 13.5 16.3 18.7 21.0DISTANCE (ft.)

1. Does the data seem reasonable? Explain.

Yes, the total distance increases, but more slowly as N increases.

2. Use your graphing calculator to construct a scatterplot of the data with Nas the input and T as the output. Using a window of Xmin � 0, Xmax � 7,Ymin � 0, and Ymax � 25, your graph should resemble the following.

3. Do you believe the data can be modeled by a logarithmic function? Explain.

Yes, because it is increasing, but more slowly as N increases.

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 380

4. This data can be modeled by log N. Use your graphing calculatorto verify visually that this is a reasonable model for the given data.

5. a. Using the log model, complete the following table. Round values to thenearest hundredth.

T � 26.75

Activity 3.11 The Elastic Ball 381

N 2 5 10

T � 26.75 log N 8.05 18.70 26.75

b. How are the outputs from 2 and 5 related to the output for 10?

T evaluated at 2 plus T evaluated at 5 equals the output at 10.

c. Using the results from part b, how could you determine the total rebounddistance after 10 bounces?

Add the result after 2 bounces and the result after 5 bounces to givethe result after 10 bounces.

The results from Problem 5 can be written as follows.

Dividing both sides by 26.75, you have

This result illustrates an important property of logarithms.

log 12 # 5 2 � log 2 � log 5.

26.75 log 12 # 5 2 � 26.75 log 2 � 26.75 log 5 26.75 log 10 � 26.75 log 2 � 26.75 log 5

26.75 � 8.05 � 18.70

Property of the Logarithm of a Product

If then where .

Expressed verbally, this property states that the logarithm of a product is the sumof the individual logarithms.

b � 1b 7 0,logb 1A # B 2 � logb A � logb B,B 7 0,A 7 0,

Example 1

a. .

b.

c.

6. Use the property of the logarithm of a product to write the following as thesum of two or more logarithms.

ln 1xy 2 � ln x � ln y

log 15st 2 � log 15 2 � log 1s 2 � log 1t 2log2 32 � log2 14 # 8 2 � log2 4 � log2 8 � 2 � 3 � 5

a. b.log3 x � log3 y � log3

z

log3 1xyz 2logb 7 � logb 13

logb 17 # 13 2

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 381

c. d.ln 3 � ln x � ln y

ln 13xy 2log 3 � log 5

log 15


7. Write the following as the logarithm of a single expression.

a. b.log4 27

log4 3 � log4 9ln 1abc 2ln a � ln b � ln c

Logarithm of a Quotient

Consider the following table from Problem 5.

N 2 5 10

T � 26.75 log N 8.05 18.70 26.75

This table also indicates that the rebound distance after this ball has hit the floortwice 18.05 ft.2 is the total rebound distance when the ball has hit the ground 10times 126.75 ft.2 minus the total rebound distance when the ball has hit the ground5 times 118.70 ft.2.This can be written as

Substituting for log 2, you have

This suggests another important property of logarithms. The property is demon-strated further in Problem 8.

8. a. Complete the following table. Round your answers to the nearestthousandth.

log 1105 2 � log 10 � log 5.

log 1105 2

log 2 � log 10 � log 5.

26.75 log 2 � 26.75 log 10 � 26.75 log 5 8.05 � 26.75 � 18.70

x Y1 � log Y2 � log x � log 4

1 �0.602 �0.602

5 0.097 0.097

10 0.398 0.398

23 0.760 0.760

(x4)

b. Is the expression equivalent to log x � log 4? Explain.

It would appear that they are. In part a, for each x, the two expres-sions produce equal outputs.

c. Sketch the graph of and using your graph-ing calculator. What do the graphs suggest about the relationship between

and log x � log 4?

The graphs are the same, indicating that the expressions are equivalent.

log 1x42y � log x � log 4y � log 1x42

log 1x42

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 382

Example 2

a.

Note that

b.

c.

9. Use the properties of logarithms to write the following as the sum or dif-ference of logarithms.

ln 1x 2

5 2 � ln 1x 2 2 � ln 15 2log 12x

y 2 � log 12x 2 � log 1y 2 � log 2 � log x � log y

log3 181272 � log3 13 2 � 1.

log3 181272 � log3 81 � log3 27 � 4 � 3 � 1


a.log6 17 � log6 3

log6 173 b.

ln x � ln 23

ln x

23

c. d.log 3 � log 2 � log z

log 32z

log3 2 � log3 x � log3 y

log3 2xy

10. Write the following expressions as the logarithm of a single expression.

a. b.log x

4z

log x � 1log 4 � log z 2log xz

4

log x � log 4 � log z

11. a. Use your graphing calculator to sketch the graphs of and

b. How do these graphs compare?

The graphs are not the same. The first is a vertical shift of The second is a horizontal shift of

c. What do the graphs suggest about the relationship between log 1A � B2and log A � log B?

log 1A � B 2 � log A � log B

y � log x.y � log x.

y � log 1x � 4 2. y � log x � log 4

Property of the Logarithm of a Quotient

If then where

Expressed verbally, this property states that the logarithm of a quotient is the dif-ference of the logarithm of the numerator and the logarithm of the denominator.

b 7 0, b � 1.logb 1AB 2 � logb A � logb B,B 7 0,A 7 0,

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 383

Logarithm of a Power

Before calculators, logarithms were used to help in computing products and quo-tients of numbers. More importantly, logarithms were used to compute powers

such as and In such a case, the first step was totake the logarithm of the power and rewrite the resulting expression. To determine

how to rewrite you can investigate the expression log

12. a. Complete the following table. Round to the nearest thousandth.

x 3.log 734.213,

20.0761 � 10.0761 21>2.734.213


x Y1 � log x3 Y2 � 3 log x

2 0.903 0.903

7 2.535 2.535

15 3.528 3.528

b. Sketch the graphs of and using your graphingcalculator.

c. What do the results of part a and part b demonstrate about the relationship

between and ?

The results in Problem 12 illustrate another property of logarithms.

log x 3 � 3 log x

3 log xlog x 3

y � 3 log xy � log x 3

Property of the Logarithm of a Power

If and p is any real number, then where In words, the property states that the logarithm of a power isequivalent to the exponent times the logarithm of the base.

b � 1.b 7 0,logb Ap � p # logb A,A 7 0

Example 3

a.

b.

c.

d.

e. log263 � log 631>2 �12 log 63

ln x 1>4 �14 ln x

ln 1xy 27 � 7 ln 1xy 2log5 x 4 � 4 log5 x

log3 92 � 2 log3 9 � 2 # 2 � 4

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 384

13. Use the properties of logarithms to write the given logarithms as the sum ordifference of two or more logarithms, or as the product of a real number anda logarithm. All variables represent positive numbers.


a.12 log3 x

log3 x 1>2 b.3 log5 x

log5 x 3

d. (Hint:13 log 50

23 50 � 501>3 2log 23 50c.2 ln t

ln t 2

e.2 log5 x � 3 log5 y � log5 z

log5 x 2y 3

z f.log3 3 � 2 log3 x � 3 log3 y

log3 3x2

y3

14. Write each of the following as the logarithm of a single expression withcoefficient 1.

b.

log 2x4

y5

12 log x 4 �

12 log y 5a.

log3 125 # 8 2 � log3 2002 log3 5 � 3 log3 2

c. d.

ln 143 2 � ln 154 # 32 2 � ln 43

54 # 32

3 ln 4 � 14 ln 5 � 2 ln 3 2log

b 5

3 # 23 # 954 � log

b 72

5

3 logb 10 � 4 logb 5 � 2 logb 3

Using the properties of logarithms to solve exponential equations algebraicallywill be investigated in the next activity.

Change of Base Formula

Because the TI-83/T1-84 Plus has only the log base 10 (log) and the log base e(ln) keys, you cannot graph a logarithmic function such as directly.Consider the following argument to rewrite the expression as an equivalentexpression using log base 10.

By definition of logs, is the same as Taking the log base 10 ofboth sides of the second equation, you have

Using the property of the log of a power, Solving for y, you have

.

Therefore, the equation is equivalent to

15. To graph , enter log 1X2 / log122 for Y1 in your TI-83/TI-84 Pluscalculator. Your graph should resemble the following.

y � log2 x

y �log xlog 2.y � log2 x

y �log xlog 2

log x � y log 2.

log x � log 2y.

x � 2y,x � 2y.y � log2 x

log2 xy � log2 x

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 385

16. a. Write as an equivalent equation using base 10.

b. Use the result from part a to graph

c. What is the domain of the function?

d. What is the x-intercept of the graph?

11, 02

x 7 0

y � log6 x.

y �log xlog 6

y � log6 x


The formula you used in problems 15 and 16 for graphing log functions ofdifferent bases is a special case of the formula

This is often called the change of base formula, where and b � 1.b 7 0

logb x �loga xloga b

, where a 7 01 � 1

The change of base formula is used to change from base b to base a. Because mostcalculators have log base 10 1log2 and log base e 1ln2 keys, you usually convert toone of those bases. For those bases,

logb x �log xlog b

or logb x �ln xln b

.

a. Using base 10: b. Using base e:

ln 11024 2ln 14 2 � 5

log 1024log 4 � 5

c. How do the results in parts a and b compare?

The results are the same.

Example 4 Change the equation y � log5 x to an equivalent equation in base10 and/or base e.

or

17. Use each of the change of base formulas to determine log4 1024.

y � log5 x �ln xln 5

y � log5 x �log xlog 5

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 386

Properties of the Logarithmic FunctionIf and then

1.

2.

3.

4.

5. You can use the calculator to change logarithms in base b to common ornatural logarithms by

or .logb x �ln xln b

logb x �log xlog b

logb Ap � p logb A

logb 1x � y 2 � logb x � logb y

logb aABb � logb A � logb B

logb 1A # B 2 � logb A � logb B

b � 1,b 7 0,B 7 0,A 7 0,




1. Use the preceding properties of logarithms to write the following as a sum ordifference of two or more logarithms.


a. b.log3 3 � log3 13 � 1 � log3 13

log3 13 # 13 2logb 3 � logb 7

logb 13 # 7 2

c.log7 13 � log7 17

log7 1317 d.

log3 x � log3 y � 1

log3 x � log3 y � log3 3 �

log3 xy3

2. Write the following expressions as the logarithm of a single number.

a. b.

log 2517

log 25 � log 17

log3 15

log3 5 � log3 3

c. d.

ln x � 7x

ln 1x � 7 2 � ln x

log5 7x5

log5 x � log5 5 � log5 7

3. a. Sketch the graphs of and on your graphingcalculator.

y � log x � log 2y � log 12x 2

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 387

b. Are you surprised by the results? Explain.

The graphs are the same. This is not surprising because the log of aproduct is the sum of the logs.

4. a. Sketch the graphs of and on your graphingcalculator.

b. Are you surprised by your results? Explain.

Not really. The graphs are reflections in the x-axis.

c. If your graphs in part a are not identical, can you modify the secondfunction to make the graphs identical? Explain.

To make them identical, change all the signs: .

5. You have been hired to handle the local newspaper advertising for a largeused car dealership in your community. The owner tells you that your prede-cessor in this position used the formula

to decide how much to spend on newspaper advertising over a 2-weekperiod. The owner admitted that he didn’t know much about the formulaexcept that N 1A2 represented the number of cars that the owner could expectto sell, and A was the amount of money that was spent on local newspaperadvertising. He also indicated that the formula seemed to work well. You canpurchase small ads in the local paper for $15 per day, larger ads for $50 perday, and giant ads for $750 per day.

a. How many cars do you expect to sell if you purchase one small ad?

, or 9 cars

b. To understand the relationship between the amount spent on advertising andthe number of cars sold, you set up a table. Complete the following table.

7.4 log 15 � 8.7

N1A 2 � 7.4 log A

y � log 3 � log x

y � log x � log 3y � log 13x 2


AD COST, A EXPECTED CARSALES, N11A22

15 9

50 13

750 21

c. How do the expected car sales from one small ad and one larger ad com-pare to the expected car sales from just one giant ad?

The sum of the sales from the smaller ads exceeds the sales from thelarger ad by 1.

DANFMC03_0321447808 pp3 12/27/06 8:12 PM Page 388

d. Are the results in the previous table above consistent with what you knowabout the properties of logarithms? Explain.

Pretty close. 15 times 50 equals 750, so I would have expected thesum of the sales from the smaller ads to equal the sales from thelargest. The error is due to rounding.

e. What are you going to advise the owner regarding the purchase of a giantad?

Forget about the giant ad. It is a waste of money.

6. Use the properties of logarithms to write the given logarithms as the sum ordifference of two or more logarithms or as the product of a real number anda logarithm. Simplify, if possible. All variables represent positive numbers.


a. b.x log2 2 � x

log2 2x

5 log3 3 � 5

log3 35

d.

13

ln x �14

ln y � 2 ln z

ln 23 x24 y

z 2c.

3 logb x � 4 logb y

logb x 3

y 4

e., it does not simplify

7. Write each of the following as the logarithm of a single expression withcoefficient 1.

log3 12x � y 2log3 12x � y 2

b.

log B4 x 3

z5

14 log x 3 �

14 log z5a.

log2 245

2 log2 7 � log2 5

c.

d.

8. Given that and that determine the numeric value ofeach of the following.

loga y � 25,loga x � 6

log5 x 2 � 3x � 2x 2 � 6x � 9

log5 1x � 2 2 � log5 1x � 1 2 � 2 log5 1x � 3 2ln

2252z4

53 � ln 4z4

5

2 ln 10 � 3 ln 5 � 4 ln z

a.12.5

loga 2y b.18

loga x 3

d.

� 12 � 25 � 1 � 36

2 loga x � loga y � loga a

loga x 2ya

c.15

3 � loga x 2

DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 389

9. Use the change of base formula and your calculator to determine a decimalapproximation of each of the following to the nearest ten thousandth.


b.0.7557

log6 215a.0.8271

log7 5

d.0.7112

log5 23 31c.1.5011

log13 47

10. The formula

gives the percentage, P, of students who could recall the important contentof a classroom presentation as a function of time, t, where t is the number ofdays that have passed since the presentation was given.

a. Sketch a graph of the function.

b. After 3 days, what percentage of the students will remember the impor-tant content of the presentation?

47.5%

c. According to the model, after how many days do only half ofthe students remember the important features of the presentation? Use agraphing approach.

2.83 days

1P � 50 2

P(t) = 95 – 30 log2t

t

P(t)

405060708090

102030

10 3 5 7 92 4 6 8 10Days since Presentation

Stu

den

ts R

ecal

ling

Imp

ort

ant

Co

nte

nt

(%)

P � f 1t 2 � 95 � 30 log2 t

DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 390

You are a criminal justice major at the local community college. The followingstatistics appeared in one of your required readings relating to the inmate popula-tion of U.S. federal prisons.

Activity 3.12 Prison Growth 391

ACTIVITY 3.12

Prison Growth

OBJECTIVE

1. Solve exponential equa-tions both graphicallyand algebraically.

YEAR 1975 1979 1986 1990 1994 1998 2000 2003

TOTAL SENTENCED POPULATION,PT (in thousands)

20.1 21.5 31.8 47.8 76.2 95.5 112.3 158.0

TOTAL SENTENCED DRUG OFFENDERS,PD (in thousands)

5.5 5.5 12.1 25.0 46.7 56.3 63.93 86.9

You decide to analyze the prison growth situation for a project in your criminologycourse.

1. Although the years in the table are not evenly spaced, you notice that each ofthe populations seems to grow rather slowly at first and more quickly later. Doyou think the data will be better modeled by linear or exponential functions?

exponential

2. Let t represent the number of years since 1970. Use your graphing calculatorto produce a scatterplot of the total inmate population, PT. Your screen shouldappear as follows.

3. Use your graphing calculator to determine the regression equation of anexponential function that best represents the total inmate population, PT.Remember that the input variable is t, the number of years since 1970. In yourregression equation, round the value for a to two decimal placesand the value for b to three decimal places. Write your model below.

4. Use your graphing calculator to visually check how well the equation in Prob-lem 3 fits the data. Using the window Xmin � �3, Xmax � 40, Ymin � �5,and Ymax � 200, your graph should resemble the following.

PT 1t 2 � 11.3611.080 2tPT � abt,

DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 391

5. Use the exponential regression model from Problem 3 to determine the totalfederal prison inmate population in 2010.

Therefore, approximately 246,800 inmates will be in federal prisonin 2010.

6. a. Using your model from Problem 3, write an equation that can be used todetermine the year in which the total federal inmate population, PT, is180,000. Remember, the population in the model is given in thousands.

b. Solve the equation in part a using a graphing approach. Your screenshould resemble the following. What is the equation of the horizontalline in the graph?

The equation of the horizontal line is In or 2006,the federal inmate population will reach 180,000.

To solve the equation for t using an algebraic approach, youneed to remove t as an exponent. The following problem guides you through thisprocess. As you will discover, logarithms are essential in this algebraic approach.

7. Solve for t using an algebraic approach.

a. Isolate the exponential factor on one side of the equation.

b. Take the log (or ln) of each side of the equation in part a.

c. Apply the appropriate property of logarithms on the left side of theequation to remove t as an exponent.

d. Solve the resulting equation in part c for t.

In 2006, there will be approximately 180,000 inmates in federalprison.

t �ln 1 180

11.362ln 11.080 2 � 35.9

t # ln 11.080 2 � ln 1 18011.362

ln 11.080 2t � ln 1 18011.362

11.080 2t �180

11.36

11.3611.080 2t � 180

11.080 2t11.3611.086 2t � 180

11.3611.080 2t � 180

1970 � 36,y � 180.

11.3611.080 2t � 180

PT 140 2 � 11.3611.080 240 � 246.8

2010 � 1970 � 40


DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 392

e. How does your solution in part d compare to the estimate obtainedgraphically in Problem 6b?

The solutions are identical.

8. You notice that over the years, the number of drug offenders seems to becomea bigger percentage of the total population.

a. Determine an exponential regression equation to model the number oftotal sentenced drug offenders, PD. Let the input variable t represent thenumber of years since 1970. In the regression equation roundthe value for a to two decimal places and the value of b to three decimalplaces.

b. Use the exponential model to predict the total number of sentenced drugoffenders in federal prisons in 2010.

The model predicts 203,100 sentenced drug offenders in 2010.

c. Write an equation that can be used to determine the year in which thetotal number of sentenced drug offenders will reach 150,000.

d. Solve the equation in part c using an algebraic approach.

The population will reach 150,000 by 2007.

Radioactive Decay

Radioactive substances, such as uranium-235, strontium-90, iodine-131, andcarbon-14, decay continuously with time. If P0 represents the original amount ofa radioactive substance, then the amount P present after a time t (usually mea-sured in years) is modeled by

where k represents the rate of continuous decay.

P � P0ekt,

t �ln 1 150

2.612ln 11.115 2 � 37.2

t # ln 11.115 2 � ln 1 1502.612

ln 11.115 2t � ln 1 1502.612

11.115 2t �1502.61

2.6111.115 2t � 150

2.6111.115 2t � 150

PD140 2 � 2.6111.115 240 � 203.1

PD1t 2 � 2.6111.115 2t

PD � abt,


DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 393

Example 1 One type of uranium decays at a rate of 0.35% per day. If 40pounds of this uranium is available today, how much will beavailable after 90 days?

SOLUTION

The uranium decays at a constant rate of 0.35% � 0.0035 per day. The initialamount, the amount available on the first day, is 40 pounds, so the equation for theamount available after t days is

.

To determine the amount available after 90 days, let t � 90. The amount available90 days from now is

P � 40e�.00351902 � 29.2 lb.

P � 40e�0.0035t


9. Strontium-90 decays continuously at a constant rate of 2.4% per year. There-fore, the equation for the amount P of strontium-90 after t years is

a. If 10 grams of strontium-90 are present initially, determine the number ofgrams present after 20 years.

g

b. How long will it take for the given quantity to decay to 2 grams?

c. How long would it take for the given amount of strontium-90 to decay toone-half of its original size (called its half-life)? Round to the nearestwhole number.

d. Do you think that the half-life of strontium-90 is 29 years regardlessof the initial amount? Answer part c using P0 as the initial amount.(Hint: Find t when )

or or yr.t � 28.9ln 10.5 2 � �0.024t12 P0 � P0e�0.024t

P �12 P0.

t � 29 yr.

ln 10.5 2�0.024

� t

ln 10.5 2 � �0.024t

0.5 � e�0.024t

510

�1010

e�0.024t

t � 67 yr.

t �ln 12 2

�0.024

ln 10.2 2 � �.024t

0.2 � e�.024t

210

�10e�.024t

10

P � 10e�0.0241202 � 6.2

P � P0e�0.024t.

DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 394


1. The number of arrests for possession of marijuana in New York City hasincreased dramatically. The number of arrests can be modeled by

where N 1t2 represents the number of arrests in thousands and t represents thenumber of years since 1990.

a. How many arrests were made in 2000?

111,700 arrests in 2000

b. According to the model, in what year were there 50,000 arrests?

Rounding up, there will be more than 50,000 arrests 9 years after 1990,

or 1999.

2. The U.S. Department of Transportation recommended that states adopt a 0.08%blood-alcohol concentration as the legal measure of drunk driving. Medicalresearch has shown that as the concentration of alcohol in the blood increases,the risk of having a car accident increases exponentially. The risk, R, expressedas a percentage, is modeled by

where x is the blood-alcohol concentration, expressed as a percent.

a. What is the risk of having a car accident if your blood-alcohol concentrationis 0.08% ?

There is a 17% risk of a car accident for a driver whose blood-alcoholconcentration is 0.08.

R10.0008 2 � 6e12.7710.082 � 17

1x � 0.08 2

R1x 2 � 6e12.77x,

t ln 11.77 2 � ln 1 500.372 t � 8.59

0.3711.77 2t � 50

N110 2 � 0.3711.77 210 � 111.7

N1t 2 � 0.3711.77 2t,


To solve exponential equations of the form where and :

1. Isolate the exponential factor on one side of the equation.

2. Take the log (or ln) of each side of the equation.

3. Apply the property log to remove the variable x as an exponent.

4. Solve the resulting equation for the variable.

b x � x log b

c 7 0b � 1,b 7 0,a 7 0,ab

x � c,SUMMARYACTIVITY 3.12


DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 395

b. What blood-alcohol concentration has a corresponding 25% risk of a caraccident?

3. In 1990, the International Panel on Climate Change projected the follow-ing future amounts of carbon dioxide (in parts per million or ppm) in theatmosphere.

x �ln 125

6 212.77

� 0.11%

12.77x � ln 1256 2

e12.77x �256

6e12.77x � 25


YEAR 1990 2000 2075 2175 2275

AMOUNT OF CARBON DIOXIDE (ppm) 353 375 590 1090 2000

a. Use your graphing calculator to create a scatterplot of the data. Let trepresent the number of years since 1990 and A1t2 represent the amountof carbon dioxide (in ppm) in the atmosphere. Do the carbon dioxidelevels appear to be growing exponentially? Yes

b. Use your graphing calculator to determine the regression equation of anexponential model that best fits the data.

c. Use the model in part b to determine in what year the 1990 carbon dioxidelevel is expected to double.

116 years after 1990 would be the year 2106.

d. Verify your result in part c graphically.

t �ln 1 705.3

352.652ln 11.006 2 � 116

t ln 11.006 2 � ln 1 705.3352.652

1.006t �705.3352.65

705.3 � 1352.65 2 11.006 2t

A1t 2 � 352.6511.006 2t

DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 396

In Exercises 4–9, solve each equation using an algebraic approach. Verify youranswers graphically.

4.

x �ln 14ln 2 � 3.81

x ln 2 � ln 14

2x � 14


6.

t �ln 2

ln 11.04 2 � 17.7

2 � 1.04t

1000 � 50011.04 2t

5.

x �ln 8

2 ln 3 � 0.95

2x ln 3 � ln 8

32x � 8

7. (Hint: Take the naturallog of both sides.)

t � 13.86

0.05t � ln 2

e0.05t � 2

8.

x � 1.881

3x � 5.6438562

13x � 1 2 �ln 1100 2

ln 12 2

23x�1 � 100 9.

t � �2.31

�0.3t � ln 2

e�0.3t � 2

10. a. Iodine-131 disintegrates at a continuous constant rate of 8.6% per day.Determine its half-life. Use the model

where t is measured in days. Round your answer to the nearest wholenumber.

b. If dairy cows eat hay containing too much iodine-131, their milk will be un-safe to drink. Suppose that hay contains 5 times the safe level of iodine-131.How many days should the hay be stored before it can be fed to dairy cows?

(Hint: Find t when )

days

11. a. In 1969 a report written by the National Academy of Sciences (U.S.) esti-mated that Earth could reasonably support a maximum world population of10 billion. The world’s population was approximately 3.6 billion and grow-ing continuously at 2% per year. If this growth rate remained constant, inwhat year would the world population reach 10 billion, referred to asEarth’s carrying capacity? Use the model

where P is the population (in billions), and t is thenumber of years since 1969.

yr.

1969 � 51 � 2020

t �ln 1 10

3.620.02

� 51

10 � 3.6e0.02t

k � 0.02,P0 � 3.6,

P � P0e kt,

15 P0 � P0

e�0.086t or t �ln 10.2 2�0.086 � 19

P �15

P0.

t � 8 days

ln 10.5 2 � �0.086t

P � P0e�0.086t,

DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 397

b. According to your growth model, when would this 1969 population double?

or ;

sometime in the year 2003

c. The world population in 1995 was approximately 5.7 billion. How does thiscompare with the population predicted by your growth model in part a?

yr.

billion

The actual population was approximately 360 million below theprediction.

d. The growth rate in 1995 was 1.5%. Assuming this growth rate remainsconstant, determine when Earth’s carrying capacity will be reached.Use the model

; the year would be 2032.t �ln 1 10

5.720.015

� 37.5

10 � 5.7e0.015t

P � P0e kt.

P 126 2 � 3.6e0.021262 � 6.06

1995 � 1969 � 26

t �ln 20.02 � 34.7e0.02t � 2


DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 398

Raising a musical note one octave has the effect of doubling the pitch, or fre-quency, of the sound. However, you do not perceive the note to sound “twice ashigh,” as you might predict. Perceived pitch is given by the function

where P is the perceived pitch in mels (units of pitch) and f is the frequency in hertz.

1. Let frequency (input) vary in value from 10 to 100,000 hertz, and let the per-ceived pitch (output) vary from 0 to 6000 mels. Graph this equation on yourgraphing calculator, using the following window: Xmin � 0, Xmax � 100,000,Ymin � 0, and Ymax � 6000.

2. What is the perceived pitch, P, for the input value 10,000 hertz?

mels

3. a. Write an equation that can be used to determine what frequency, f, givesan output value of 2000 mels.

b. Solve the equation in part a using a graphing approach.

The solution is 3599 hertz.

To determine the exact answer in Problem 3, you can use an algebraic approach.The following problem guides you through this process.

4. Solve the equation using an algebraicapproach.

a. Solve the equation for That is, isolate the log on oneside of the equation.

b. The equation in part a is now in the form where , and Write the equation from part a in

exponential form,

0.0016f � 1 � 10 2000>2410

bE � N.E �

20002410.N � 0.0016x � 1

b � 10,logb N � E,

log 10.0016f � 1 2 �20002410

log 10.0016f � 1 2.

2410 log 10.0016f � 1 2 � 2000

2410 log 10.0016f � 1 2 � 2000

P 110,000 2 � 2410 log 10.0016110,000 2 � 1 2 � 2965.38

P1 f 2 � 2410 log 10.0016 f � 1 2,

Activity 3.13 Frequency and Pitch 399

ACTIVITY 3.13

Frequency and Pitch

OBJECTIVE

1. Solve logarithmic equa-tions both graphicallyand algebraically.

DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 399

c. In exponential form, the equation in part a should be

Solve this equation for f. Of course, you will need to approximate a valueof using your calculator.

hertz

d. How does your answer to part c compare to your answer to Problem 3b?

The answers are the same.

5. a. Use an algebraic approach to determine the frequency, f, that produces aperceived pitch of 3000 mels.

Hz

b. Verify your answer in part a using a graphing approach.

6. The formula is a model for the average walking speed,W, in feet per second for a resident of a city with population P, measured inthousands.

a. Determine the walking speed of a resident of a small city having apopulation of 500,000.

ft./sec.

b. If the average walking speed of a resident is 4.5 feet per second, what isthe population of the city? Round your answer to the nearest thousand.

the population is 153,000.e14.5�2.742> 0.35 � P � 153;

4.5 � 0.35 ln P � 2.74

W � 0.35 ln 1500 2 � 2.74 � 4.92

W � 0.35 ln P � 2.74

1103000>2410 � 1 20.0016

� f � 10,357.30

3000 � 2410 log 10.0016f � 1 2

f �102000>2410 � 1

0.0016� 3599

102000>2410

0.0016 f � 1 � 102000>2410.


To solve a logarithmic equation algebraically,

Step 1. Rewrite the equation in the form where and

Step 2. Rewrite the resulting equation from step 1 in exponential form,

Step 3. Solve the resulting equation from step 2 algebraically.

Step 4. Check the solutions in the original equation.

f 1x 2 � bc.

f 1x 2 7 0.c 7 0,b � 1,b 7 0,logb 1 f 1x 2 2 � c,


DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 400


In Exercises 1–6, solve each equation using an algebraic approach. Then verifyyour answer using a graphical approach.

1.x � 25 � 32

log2 x � 5


Activity 3.13 Frequency and Pitch 401

2.x � e10 � 22,026

ln x � 10

3.

x � 55>3 � 2 � 12.62

x � 2 � 55>33 log5 1x � 2 2 � 5 4.

x � 29

x � 4 � 52

log5 1x � 4 2 � 2

5.

x � 303.17

ln x � 5.7143

20 � 3.5 ln x 6.

x � 5,018,148

ln x � 15.4286

4 � 1.75 ln x � 31

7. Stars have been classified into magnitude according to their brightness. Starsin the first six magnitudes are visible to the naked eye; those of highermagnitudes are visible only through a telescope. The magnitude, m, of thefaintest star that is visible with a telescope having lens diameter d, in inches,is modeled by

What is the highest magnitude of a star that is visible with the 200-inch telescopeat Mount Palomar, California?

8. Coal consumption in the United States can be modeled by the equation

where x is the number of years since 1970, and A1x2 is the amount of coal con-sumed in quadrillions of British thermal units or quads. According to themodel, in what year will the consumption of coal in the United States reach30 quads?

The year is 1970 � 214 � 2184.

x � 214

5.36 � ln x

30 � 4.95 � 4.67 ln x

A1x 2 � 4.95 � 4.67 ln x,

m � 8.8 � 5.1 log 200 � 20.5

m � 8.8 � 5.1 log d.

DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 401

9. The acidity or alkalinity of any solution is determined by the concentrationof hydrogen ions, [H�], in the substance, measured in moles per liter (mol/l).Acidity (or alkalinity) is measured on a pH scale, using the model

The pH scale ranges from 0 to 14. Values below 7 have progressively greateracidity; values greater than 7 are progressively more alkaline. Normal un-polluted rain has a pH of about 5.6. The acidity of rain over the northeasternUnited States, caused primarily by sulfur dioxide emissions, has had verydamaging effects. One of the most acidic rainfalls on record had a pH of 2.4.What was the concentration of hydrogen ions?

mol/l

10. The Richter scale is a well-known method of measuring the magnitude of anearthquake in terms of the amplitude, A (height), of its shock waves. Themagnitude of any given earthquake is given by

where A0 is a constant representing the amplitude of an average earthquake.

a. The magnitude of the 1906 San Francisco earthquake was 8.3 on theRichter scale. Write an equation that gives the amplitude, A, of the SanFrancisco earthquake in terms of A0.

b. An earthquake with a magnitude of 5.5 will begin to cause serious damage.Write an equation that gives the amplitude, A, of a serious-damage earth-quake in terms of A0.

c. Determine the ratio of the amplitude of the San Francisco earthquake tothe amplitude of a serious-damage (magnitude 5.5) earthquake. What isthe significance of this number?

The amplitude of the San Francisco earthquake was 631 times greaterthan that of the smallest earthquake that causes serious damage.

A0108.3

A0105.5 � 102.8 � 631

A � A0105.5

A � A0# 108.3

108.3 �AA0

8.3 � log 1 AA02

m � log 1 AA02 ,

3H� 4 � 10�2.4 � 0.00398

2.4 � �log 3H� 4

pH � �log 3H� 4 .


DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 402

1. A logarithm is an exponent. Explain how this fact relates to the followingproperties of logarithms.

a.Multiplication of powers with the same base is accomplished byadding the exponents.

b.Division of powers of the same base is accomplished by subtractingthe exponents.

c.Exponentiation of a power is accomplished by multiplying theexponents.

2. You have $20,000 to invest. Your broker tells you that the value of shares ofmutual fund A has been growing exponentially for the past 2 years and thatshares of mutual fund B have been growing logarithmically over the sameperiod. If you make your decision based solely on the past performances ofthe funds, in which fund would you choose to invest? Explain.

I will choose fund A, because exponential growth results in morerapid growth over time. Logarithmic growth results in slower growthover time.

3. Study the following graphs showing various types of functions you haveencountered in this course.

a. b.

c. d. y

x

y

x

y

x

y

x

logb x n � n # logb x

logb xy � logb x � logb y

logb 1x # y 2 � logb x � logb y

Cluster 2 What Have I Learned? 403

CLUSTER 2 What Have I Learned?

DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 403

e. f.

Complete the following table with respect to the preceding graphs.

y

x

y

x


Constant function b

Linearly decreasing function c ;

Logarithmically increasing function d ;

Exponentially decreasing function e ;

Exponentially increasing function a ;

Linearly increasing function f ; m 7 0y � mx � b

b 7 1y � bx

0 6 b 6 1y � bx

b 7 1y � logb x

m 6 0y � mx � b

y � a

DESCRIPTION GRAPH LETTER GENERAL EQUATION

4. The graph of will never be located in the second or third quadrants.Explain.

The domain of is , since by cannot be negative.

5. What function would you enter into Y1 on your graphing calculator to graphthe function

6. What values of x cannot be inputs in the function ?

7. What is the relationship between the functions and ? Howare the graphs related?

The functions are inverses. The graphs are symmetric through the liney � x.

y � 10 xy � log x

3x � 2 � 0 or x �23

y � logb 13x � 2 2

y �log xlog 4

y � log4 x?

x 7 0y � logb x

y � logb x

DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 404

1. Write each equation in logarithmic form.


CLUSTER 2 How Can I Practice?


a. b. c.

log3 1 1812 � �4

3�4 �181

log10 10.0001 2 � �4

0.0001 � 10�4

log4 16 � 2

42 � 16

2. Write each equation in exponential form.

a.25 � 32

log2 32 � 5 b.50 � 1

log5 1 � 0

c.

10�3 � 0.001

log10 0.001 � �3 d.e1 � e

ln e � 1

3. Solve each equation for the unknown variable.

a. b. c.

y � 3

5y � 125 � 53

log5 125 � y

b � 2

b5 � 32 � 25

logb 32 � 5

x � 4�3 �164

log4 x � �3

4. a. Complete the table of values for the function f 1x 2 � log4 x.

x 0.25 0.5 1 4 16 64

f 11x22 �1 �0.5 0 1 2 3

b. Sketch a graph of the function, f.

4 8

–1

–2

1

2

3

12 16 20

f(x) = log4x

y

x

DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 405

c. Use your graphing calculator to check your result in parts a and b.

d. Determine the x-intercept.

11, 02

e. What is the domain of the function?

f. What is the range?

all real numbers

g. Does the graph have a vertical or horizontal asymptote?

The y-axis is a vertical asymptote.

h. Use your graphing calculator to determine f 1322.

i. Use your graphing calculator to determine x when

5. Write each of the following as a sum, difference, or multiple of logarithms.Assume that x, y, and z are all greater than 0.

a.

logb x � 2 logb y � logb z

logb xy 2

z

x � 90.5

f 1x 2 � 3.25.

f 132 2 � 2.5

1x � 0 2

x 7 0


b.32 log3 x �

12 log3 y � log3 z

log3 2x 3yz

c.log5 x �

12 log5 1x 2 � 4 2

log5 1x2x 2 � 4 2 d.13 log4 x �

23 log4 y �

23 log4 z

log4 33 xy 2

z2

6. Rewrite the following as the logarithm of a single quantity.

a.

log x23 y2z

log x �13 log y �

12 log z b.

log31x � 3 23 z2

3 log3 1x � 3 2 � 2 log3 z

c.

log3 B3 xy 2z

4

13 log3 x �

23 log3 y �

43 log3 z

7. Use the change of base formula and your calculator to approximate thefollowing.

a.log 17log 5 � 1.76

log5 17 b.13

# log 41log 13 � 0.4826

log13 23 41

DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 406

8. Solve each of the following using an algebraic approach.

a.

x � 0.0067

x � e�5

ln x � �5

3 ln x � �15

25 � 3 ln x � 10


b.

x � 646.08

x � 47>1.5 � 1

x � 1 � 47>1.5

1.5 log4 1x � 1 2 � 7

9. Solve the following algebraically. Check your solutions using graphs or tables.

a.

x �log 17log 3 � 2.5789

3x � 17 b.

x �ln 141.7 � 1.55

1.7x � ln 14

14 � e1.7x

42 � 3e1.7x

10. The following table shows per capita health care expenditures (in dollars) inthe United States from 1988 to 1993.

YEAR 1988 1989 1990 1991 1992 1993

EXPENDITURE ($) 2201 2422 2688 2902 3144 3331

a. Plot these points on the following grid.

b. The logarithmic model that fits this data is where E represents the per capita health care expenditures and x is thenumber of years since 1987. Add a sketch of this model to the grid inpart a.

c. Use this model to predict the per capita health care expenditures in 2005.

d. Use the model in part c to predict in what year health care expenditureswill reach $3500. Use a graphing approach.

which you round up to 10. The year is 1997.x � 9.38,

E � f 118 2 � 2090 � 630 ln 18 � 3910.93

E � f 1x 2 � 2090 � 630 ln x,

15002000250030003500 f(x) = 2090 + 630 lnx

5001000

1988 1989 1990 1991 1992 1993x

Year

Per

Cap

ita

Hea

lth

Car

e S

pen

din

g (

$)

y

DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 407

e. Write the equation that you would solve to determine the answer to part d.

f. Solve the equation in part e algebraically, and compare your result withyour answer in part d.

The year is 1997; it is the same.

x � e2.238 � 9.38

ln x � 2.238

630 ln x � 1410

2090 � 630 ln x � 3500

2090 � 630 ln x � 3500


DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 408

409

SummaryC h a p t e r 3

The bracketed numbers following each concept indicate the activity in which the concept is discussed.

CONCEPT / SKILL DESCRIPTION EXAMPLE

Exponential functions [3.1] The exponential functions are defined by.b � 1y � b

x, b 7 0,y � 3x

Horizontal asymptote of anexponential function [3.1]

The line is a horizontal asymptoteof an exponential function .y � b

xy � 0 As x gets smaller, the output

values of approach 0.y � 3x

Vertical intercept of an exponentialfunction [3.1]

The vertical intercept (y-intercept) of anexponential function is 10, 12.y � b

xThe graph of passesthrough the point 10, 12.y � 2x

Growth factor of an exponentialfunction [3.1]

If , the function y � bx is increas-ing and b is called the growth factor.

b 7 1 The exponential function has a growth factor of 3.

y � 3x

Decay factor of an exponentialfunction [3.1]

The exponential function has a decay factor of 12.

y � 1122x

Half-life [3.2] The half-life of a decreasing exponentialfunction is the time it takes for an outputto decay by one-half. The half-life isdetermined by the decay factor andremains the same for all output values.

Example 3, Activity 3.2;see pages 302–303

Doubling time [3.2] The doubling time of an increasingexponential function is the time it takesfor an output to double. The doublingtime is set by the growth factor andremains the same for all output values.

Example 2, Activity 3.2; seepages 300–301

Decay model [3.3] If r represents the annual percent thatdecays, the exponential function thatmodels the amount remaining can bewritten as , where

is the decay factor.1 � rP1t 2 � P011 � r 2t

Example 5, Activity 3.3;see page 313

Compound interest [3.5] The formula for compounding interest is.A � P 11 �

rn2nt


Growth model [3.3] If r represents the annual percentagegrowth rate, the exponential functionthat models the quantity P can bewritten as , where P0is the initial amount, t represents thenumber of elapsed years, and isthe growth factor.

1 � r

P1t 2 � P011 � r 2t


If , the function y � bx is de-creasing and b is called the decay factor.

0 6 b 6 1

DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 409


Common logarithm [3.8] A common logarithm is a base-10 loga-rithm. The notation is .log10 x � log x

. The commonlogarithm of 103 is 3; i.e.,

.log 1000 � 3

1000 � 103

Logarithmic function [3.9] If and , the logarithmicfunction is defined by .y � logb x

b � 1b 7 0 y � log4 x

Basic properties of logarithms [3.8] If and , ,, and logb bn � n.logb b � 1

logb 1 � 0b � 1b 7 0 , ,log6 64 � 4

log7 7 � 1log4 1 � 0

Comparison of the graphs of f 1x2� bx, where b 1, and g 1x2 � logb 1x2, where b 1 [3.9]

Problem 6, Activity 3.9;see page 368

Graph of the logarithmicfunction [3.9]

The graph is increasing for all ,has an x-intercept of 11, 02 and has avertical asymptote of , the y-axis.x � 0

x 7 0 y

x

Natural logarithm [3.8] A natural logarithm is a base-e logarithm.The notation is .loge x � ln x

loge e3 � ln e3 � 3

Logarithmic equation [3.8] The logarithmic equation isequivalent to the exponential equation

.b y � x

y � logb x The equations andare equivalent.x � 46

6 � log4 x

Continuous compounding [3.5] The formula for continuous compound-ing is .A � Pert


Continuous decay at a constantpercentage rate [3.6]

Whenever decay is continuous at a con-stant rate, the model used is .y � y0e�rt


Notation for logarithms [3.8] The notation for logarithms is ,where b is the base of the log, x (a posi-tive number) is the power of b, and y isthe exponent, and x is the power of b.

logb x � y In the equation , 2 isthe base, 4 is the log or exponent,and 16 is the power of z.

log2 16 � 4

Continuous growth at a constantpercentage rate [3.5], [3.6]

Whenever growth is continuous at aconstant percentage rate, the exponentialmodel used is .y � y0ert

Problem 10, Activity 3.5;see page 325

Logarithm [3.8] In the equation , where and, x is called a logarithm or log.b � 1

b 7 0y � b x For the equation 4 is the

logarithm of 81, base 3.34 � 81,

410

Both graphs increase. The exponentialfunction increases faster as x increases;the log function increases slower as xincreases. The domain of the exponentialfunction is the range of the log, which isall real numbers; the range of theexponential function is the domain of thelog, which is the interval 10, 2.

DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 410


411

If A 0, B 0, b 0, and b � 1,then logb 1A B2� logb A � logb B[3.11]

#The logarithm of a product is the sum ofthe logarithms. � 2 � 3 � 5

log2 14 # 8 2 � log2 14 2 � log2 18 2

If A 0, B 0, b 0, and b � 1,then logb 1A � B2� logb 1A2 �logb 1B2 [3.11]

The logarithm of a sum is not the sum ofthe logarithms.

log 12 � 3 2 � log 5 � 0.69900.3010 � 0.4771 � 0.7781log 2 � log 3 �

Change of base formula [3.11] The logarithm of any positive numberx to any base can be found using theformula

or logb x �ln xln b

.logb x �log xlog b

log2 12.5 2 �log 12.5 2

log 2� 1.3219

If A 0, B 0, b 0, and b � 1,then logb � logb A � logb B[3.11]

1AB2The logarithm of a quotient is the dif-ference of the logarithms.

� 4 � 3 � 1

log3 81 � log3 27log3 181272 �

If A 0, p is a real number, b 0,and b � 1, then logb Ap� p logb A[3.11]

The logarithm of a power of A is theexponent times the logarithm of A.

log3 2x �12

log3 x

log5 x 4 � 4 log5 x

DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 411

DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 412

1. a. Determine some of the output values for the function by completingthe following table.

f 1x 2 � 8x

413

Gateway Review

Answers to all Gateway exercises are included in the Selected Answers appendix.

x �1 0 1 2 3

1 8 16 64 51212

18f (x) � 8x

43�

13

b. Sketch the graph of the function, f.

c. Is this function increasing or decreasing? Explain how you know this by lookingat the equation of the function.

The function is increasing, because .

d. What is the domain?

all real numbers

e. What is the range?

f. What are the x- and y-intercepts?

There is no x-intercept. The y-intercept is 10, 12.g. Are there any asymptotes? If yes, write the equations of the asymptotes.

There is one horizontal asymptote, the x-axis, .

h. Compare the graph of f to the graph of . What are the similaritiesand the differences?

The domain and range are the same. The graphs are reflections in the y-axis. f is increasing; g is decreasing.

i. In what way does the graph of differ from that of ?

f is moved upward 5 units to obtain h.

f 1x 2 � 8xh1x 2 � 8x � 5

g1x 2 � 1182xy � 0

y 7 0

b � 8 7 1

f(x) = 8x

2 31–1

30

50

10

70

f(x)

x

C h a p t e r 3

DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 413

j. Write the equation of the function that is the inverse of the function f 1x2.Interchange x and y: ; solve for y:

2. Complete the table for each exponential function. Use your graphing calculator tocheck your work.

y � log8 xx � 8y

414

6 growth none 10, 12 increasing

decay none 10, 12 decreasing

2.34 growth none 10, 52 increasing

0.78 decay none 10, 32 decreasing

2 growth 12, 02 10, �32 increasingy � �4r 1x 2 � 2x � 4

y � 0q1x 2 � 310.78 2xy � 0p1x 2 � 512.34 2xy � 01

3g1x 2 � 1132xy � 0h1x 2 � 6x


3. Use your graphing calculator to help you determine the domain and range for eachfunction.

FUNCTION

DOMAIN all reals all reals all reals

RANGE all reals all realsy 7 �5y 7 2y 7 0

x 7 3x 7 0

r 1x 2 � ln 1x � 3 2q1x 2 � log4 xt1x 2 � 3x � 5h1x 2 � 6x � 2f 1x 2 � 0.8x

4. a. Given the following table, determine whether the given data can be approxi-mately modeled by an exponential function. If it can, what is the growth or decayfactor?

x 0 1 2 3 4

y 10 15.5 24 36 55.5

The table is approximately exponential. The growth factor is about 1.55.

b. Determine an exponential equation that models this data.

5. a. Your salary has increased at the rate of 1.5% annually for the past 5 years, andyour boss projects this will remain unchanged for the next 5 years. You weremaking $15,000 annually in 2002. Complete the following table.

y � 10 # 1.55x

2002 2003 2004 2005 2006 2007

15,000 15,225 15,453 15,685 15,920 16,159

b. Write the exponential growth function that models your annual salary duringthis period of time. Let x represent the number of years since 2002.

y � 15,00011.015 2x

DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 414

c. If your increase in salary continues at this rate, how much will you make in2010? Is this realistic?

; this is reasonable if you assume that 15,000 is a reasonablestarting salary and that the 1.5% salary increase per year remains constant.

d. You would like to double your salary. How many years will you have to workbefore your salary will be twice the salary you made in 2002?

yr.

6. a. You just inherited $5000. You can invest the money at a rate of 6.5% com-pounded continuously. In 8 years, your oldest child will be going to college.How much will be in the bank for her education? Use the equation

b. You actually need to have $12,000 for your child’s first year of college. Forhow many years would you have to leave the money in the bank to have the$12,000?

7. The number of multiple births (triplets or higher) in the United States between1990 and 1999 is listed in the following table, with 0 representing the year 1990.

t � 13.5 yr.

0.065t � ln 1125 2

12,000 � 5000e0.065t

A � 5000e0.065182 � $8410.14

A � A0ert.

x � log1.015 2 �ln 12 2

ln 11.015 2 � 46.6

y � $16,897

415

NUMBER OF YEARS 0 2 3 4 5 6 7 8 9SINCE 1990

NUMBER OF MULTIPLE 3028 3883 4168 4594 4973 5939 6737 7625 7321BIRTHS

a. Plot the data on an appropriately scaled and labeled coordinate axis.

2000

0

4000

6000

8000

10,000

0 1 2 3 4 5 6 7 8 9Years since 1990

Nu

mb

er o

f M

ult

iple

Bir

ths

Source: National Center for Health Statistics.

DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 415

b. Does the scatterplot show that the data would be better modeled by a linear oran exponential model?

It would be better modeled by an exponential model.

c. Use your graphing calculator to determine the exponential regression equationthat best fits the multiple births data.

d. According to your model, what is the growth factor for the multiple births data?

1.112

e. Estimate the growth rate (written as a percent) in multiple births each year.

11.2%

f. Use the regression equation to determine the predicted total number of multiplebirths in 2012.

g. Use your graphing calculator to determine the doubling time for your exponen-tial model.


8. Determine the value of each of the following without using your calculator.

f 122 2 � 31,635

f 1x 2 � 306111.112 2x

416

a.125

b. 813/4

27

c. 64�5/6

132

25 3>2

d.25

e.�2

log3 1923 1252 f.4

log5 625

h.2

ln e2g.�3

log 0.001

9. Write each equation in logarithmic form.

a.log6 36 � 2

62 � 36 b.log10 0.000001 � �6

0.000001 � 10�6 c.

log2 132 � �5

2�5 �1

32

10. Write each equation in exponential form.

a. b. c.

10�4 � 0.0001

log10 0.0001 � �4

70 � 1

log7 1 � 0

34 � 81

log3 81 � 4

d. e.

qb � y

logq y � b

e1 � e

ln e � 1

DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 416

11. Solve each equation for the unknown variable.

a. b.

b � 4

b4 � 256

logb 256 � 4

x �1

125

5�3 � x

log5 x � �3

d.

x � 23 � 8

x � 43>2log4 x �

32

c.

y � 6

2y � 64

log2 64 � y

12. a. Complete the table of values for the function .f 1x 2 � log5 x

x 0.008 0.04 0.2 1 5 25

f 11x22 �3 �2 �1 0 1 2

b. Sketch a graph of the function.

c. Use your graphing calculator to check your result in parts a and b.

d. Determine the x-intercept.

11, 02e. What is the domain of the function?

f. What is the range?

all real numbers

x 7 0

5 10

–1

–2

1

2

3

15 20 25

f(x) = log5 x

y

x

417

DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 417

g. Does the graph have a vertical or horizontal asymptote?

It has a vertical asymptote at . The function gets closer and closer tothe y-axis but does not cross it.

h. Use your graphing calculator to determine f 1232.

i. Use your graphing calculator to determine x when .

13. Use the change of base formula and your calculator to approximate the following.

x � 52.416

f 1x 2 � 2.46

f 123 2 � 1.948

x � 0

418

b.

log 1892log 15

� �0.0435

log15 89a.log 21log 7 � 1.56

log7 21

14. Write each of the following as a sum, difference, or multiple of logarithms.Assume that x, y, and z are all greater than 0.

b.

1132 14 log x � 3 log y � log z 2log B3 x 4y 3

za.

3 log2 x � log2 y � 1122 log2 z

log2 x 3y

z 1>2

15. Rewrite the following as the logarithm of a single quantity.

a. b.

log B3 xy 2z

13 1log x � 2 log y � log z2

log x24 y

z 3

log x �14 log y � 3 log z

16. Solve the following algebraically.

a.

; x � �1.233 � x �log 7log 3

33�x � 7

c.

17. a. Sketch the graph of the function using the data from the given table.

x � 341.5

x � e35>6 ln x �

356

6 ln x � 35

50 � 6 ln x � 85

b.

; x � 1.754x � 9 � 24

log2 14x � 9 2 � 4

x 0.1 0.5 1 2 4 16

f 11x22 �1.66 �0.5 0 .5 1 2

DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 418

b. Use the table and the graphing feature of your calculator to verify that theequation that defines function f is .

c. Use the function to determine the value of f 1542.2.87744

d. If , determine the value of x.

e. Use your graphing calculator to verify that the function is theinverse of f.

18. The population (in millions) of New York State and Florida can be modeled by thefollowing:

New York State:

Florida:

where t represents the number of years since 2000.

a. Determine the population of New York and Florida in 2000 1 2.New York 18.98 million Florida 15.98 million

b. Sketch a graph of each function on the same coordinate axes.

PF = 15.98e0.0235t

PN = 18.98e0.0055t

10

20

30

40

5–5 0 10 15 20 25t

Years since 2000

Po

pu

lati

on

(in

mill

ion

s)

P

t � 0

PF � 15.98e0.0235t

PN � 18.98e0.0055t

g1x 2 � 4x

x � 24.92.319 �log x

2 log 2;

f 1x 2 � 2.319

f 1x 2 � 0.5 log2 x

8 12 164

2

1

f(x)

x

419

DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 419

c. Determine graphically the year when the population of Florida will equal thepopulation of New York State.

Florida’s population will equal that of New York sometime in the year2009.

d. Determine algebraically the year when the population of Florida will firstexceed 25 million.

Florida’s population will exceed 25 million in the year 2019.

t � 19 yr.

ln 2515.98 � 0.0235t

25 � 15.98e0.0235t

420

DANFMC03_0321447808 pp3 12/27/06 8:13 PM Page 420

exponential and logarithmic functions - the eyethe-eye.eu/public/worldtracker.org/college...

Documents