exponential and logarithmic functions - … · 2012-12-31 · 466 appendix b exponential and...

Mathematics is the abstract key which turns the lock of the physical universe.

John Polkinghorne

If you have had any problems with or had testing done on your thyroid gland, thenyou may have come in contact with radioactive iodine-131. Like all radioactive ele-ments, iodine-131 decays naturally. The half-life of iodine-131 is 8 days, whichmeans that every 8 days a sample of iodine-131 will decrease to half of its originalamount. Table 1 and the graph in Figure 1 show what happens to a 1,600-microgramsample of iodine-131 over time.

A P P E N D I X BEXPONENTIAL ANDLOGARITHMIC FUNCTIONS

464

INTRODUCTION

0 5 10 15 20 25 30 350

500

1,000

1,500

2,000

t

A(t)

Days

Mic

rogr

ams

Figure 1

TABLE 1 Iodine-131 as a Function of Time

t A(Days) (Micrograms)

0 1,600

8 800

16 400

24 200

32 100

The function represented by the information in the table and Figure 1 is

A(t) � 1,600 � 2�t/8

It is one of the types of functions we will study in this appendix.

SECTION B.1 EXPONENTIAL FUNCTIONS

To obtain an intuitive idea of how exponential functions behave, we can consider theheights attained by a bouncing ball. When a ball used in the game of racquetball isdropped from any height, the first bounce will reach a height that is �

23

� of the originalheight. The second bounce will reach �

23

� of the height of the first bounce, and so on,as shown in Figure 1 on the following page.

If the ball is dropped initially from a height of 1 meter, then during the firstbounce it will reach a height of �

23

� meter. The height of the second bounce will reach�23

� of the height reached on the first bounce. The maximum height of any bounce is �23

�

of the height of the previous bounce.

Initial height: h � 1

Bounce 1: h � (1) � �

Bounce 2: h � � � � � �2

Bounce 3: h � � �2

� � �3

Bounce 4: h � � �3

� � �4

. .

. .

. .

Bounce n: h � � �n�1

� � �n

This last equation is exponential in form. We classify all exponential functionstogether with the following definition.

2�3

2�3

2�3

2�3

2�3

2�3

2�3

2�3

2�3

2�3

2�3

2�3

2�3

2�3

465Section B.1 Exponential Functions

Bounce1

Bounce2

Bounce3

Bounce4

h

h•23

23

h23

Figure 1

DEFINITION

An exponential function is any function that can be written in the form

f(x) � bx

where b is a positive real number other than 1.

Each of the following is an exponential function:

f(x) � 2x y � 3x f(x) � � �x

The first step in becoming familiar with exponential functions in general is tofind some values for specific exponential functions.

1�4

If the exponential functions f and g are defined by

f(x) � 2x and g(x) � 3x

then

f(0) � 20 � 1 g(0) � 30 � 1

f(1) � 21 � 2 g(1) � 31 � 3

f(2) � 22 � 4 g(2) � 32 � 9

f(3) � 23 � 8 g(3) � 33 � 27

f(�1) � 2�1 � g(�1) � 3�1 �

f(�2) � 2�2 � � g(�2) � 3�2 � �

f(�3) � 2�3 � � g(�3) � 3�3 � �

In the introduction to this appendix we indicated that the half-life of iodine-131 is8 days, which means that every 8 days a sample of iodine-131 will decrease to halfof its original amount. If we start with A0 micrograms of iodine-131, then after t daysthe sample will contain

A(t) � A0 � 2�t/8

micrograms of iodine-131.

A patient is administered a 1,200-microgram dose ofiodine-131. How much iodine-131 will be in the patient’s system after 10 days andafter 16 days?

SOLUTION The initial amount of iodine-131 is A0 � 1,200, so the function thatgives the amount left in the patient’s system after t days is

A(t) � 1,200 � 2�t/8

After 10 days, the amount left in the patient’s system is

A(10) � 1,200 � 2�10/8

� 1,200 � 2�1.25

� 504.5 mg

After 16 days, the amount left in the patient’s system is

A(16) � 1,200 � 2�16/8

� 1,200 � 2�2

� 300 mg

We will now turn our attention to the graphs of exponential functions. Becausethe notation y is easier to use when graphing, and y � f(x), for convenience we willwrite the exponential functions as

y � bx

1�27

1�33

1�8

1�23

1�9

1�32

1�4

1�22

1�3

1�2

466 Appendix B Exponential and Logarithmic Functions

EXAMPLE 1

EXAMPLE 2

Sketch the graph of the exponential function y � 2x.

SOLUTION Using the results of Example 1, we have the following table. Graph-ing the ordered pairs given in the table and connecting them with a smooth curve, wehave the graph of y � 2x shown in Figure 2.


EXAMPLE 3

x y

�3 �18

�

�2 �14

�

�1 �12

�

0 1

1 2

2 4

3 8

–5 –4 –3 –2 –1 1 2 3 4 5

–5–4–3–2–1

12

45

x

y

3y = 2x

Figure 2

Notice that the graph does not cross the x-axis. It approaches the x-axis—in fact,we can get it as close to the x-axis as we want without it actually intersecting the x-axis. For the graph of y � 2x to intersect the x-axis, we would have to find a valueof x that would make 2x � 0. Because no such value of x exists, the graph of y � 2x

cannot intersect the x-axis.

Sketch the graph of y � � �x.

SOLUTION The table shown here gives some ordered pairs that satisfy the equa-tion. Using the ordered pairs from the table, we have the graph shown in Figure 3.

1�3

EXAMPLE 4

x y

�3 27

�2 9

�1 3

0 1

1 �13

�

2 �19

�

3 �217�

–5 –4 –3 –2 –1 1 2 3 4 5

–5–4–3–2–1

12

45

x

y

3y = ( )x1

3

Figure 3

The graphs of all exponential functions have two things in common: (1) eachcrosses the y-axis at (0, 1) because b0 � 1; and (2) none can cross the x-axis because bx � 0 is impossible due to the restrictions on b.

Figures 4 and 5 show some families of exponential curves to help you becomemore familiar with them on an intuitive level.


–3 –2 –1 1 2 3–1

1

2

3

4

5

x

yy = 2x

y = 3x

y = 4x

Figure 4

x

–1

1

2

3

4

5

y

y = ( )x14

y = ( )x13

y = ( )x12

–3 –2 –1 1 2 3

Figure 5

Among the many applications of exponential functions are the applications hav-ing to do with interest-bearing accounts. Here are the details.

COMPOUND INTEREST

If P dollars are deposited in an account with annual interest rate r, compoundedn times per year, then the amount of money in the account after t years is givenby the formula

A(t) � P�1 � �ntr

�n

Suppose you deposit $500 in an account with an annualinterest rate of 8% compounded quarterly. Find an equation that gives the amount ofmoney in the account after t years. Then find the amount of money in the accountafter 5 years.

SOLUTION First we note that P � 500 and r � 0.08. Interest that is com-pounded quarterly is compounded 4 times a year, giving us n � 4. Substituting thesenumbers into the preceding formula, we have our function

A(t) � 500 �1 � �4t

� 500(1.02)4t

To find the amount after 5 years, we let t � 5:

A(5) � 500(1.02)4�5 � 500(1.02)20 � $742.97

Our answer is found on a calculator and then rounded to the nearest cent.

0.08�

4

EXAMPLE 5

The Natural Exponential Function

A commonly occurring exponential function is based on a special number we denotewith the letter e. The number e is a number like p. It is irrational and occurs in manyformulas that describe the world around us. Like p, it can be approximated with adecimal number. Whereas p is approximately 3.1416, e is approximately 2.7183.(If you have a calculator with a key labeled , you can use it to find e1 to find amore accurate approximation to e.) We cannot give a more precise definition of thenumber e without using some of the topics taught in calculus. For the work we aregoing to do with the number e, we only need to know that it is an irrational numberthat is approximately 2.7183.

Table 1 and Figure 6 show some values and the graph for the natural exponentialfunction.

y � f(x) � ex

ex


–3 –2 –1 1 2 3–1

1

2

3

4

5

x

y

y = ex

Figure 6

TABLE 1

x f (x) � ex

�2 f (�2) � e�2 � � 0.135

�1 f (�1) � e�1 � � 0.368

0 f (0) � e0 � 1

1 f (1) � e1 � e � 2.72

2 f (2) � e2 � 7.39

3 f (3) � e3 � 20.1

1�e

1�e2

One common application of natural exponential functions is with interest-bearing accounts. In Example 5 we worked with the formula

A � P�1 � �nt

which gives the amount of money in an account if P dollars are deposited for t yearsat annual interest rate r, compounded n times per year. In Example 5 the number ofcompounding periods was 4. What would happen if we let the number of com-pounding periods become larger and larger, so that we compounded the interest everyday, then every hour, then every second, and so on? If we take this as far as it can go,we end up compounding the interest every moment. When this happens, we have anaccount with interest that is compounded continuously, and the amount of money insuch an account depends on the number e.

r�n

CONTINUOUSLY COMPOUNDED INTEREST

If P dollars are deposited in an account with annual interest rate r, compoundedcontinuously, then the amount of money in the account after t years is given bythe formula

A(t) � Pert

Suppose you deposit $500 in an account with an annualinterest rate of 8% compounded continuously. Find an equation that gives the amountof money in the account after t years. Then find the amount of money in the accountafter 5 years.

SOLUTION The interest is compounded continuously, so we use the formulaA(t) � Pert. Substituting P � 500 and r � 0.08 into this formula we have

A(t) � 500e0.08t

After 5 years, this account will contain

A(5) � 500e0.08�5 � 500e0.4 � $745.91

to the nearest cent. Compare this result with the answer to Example 5.


EXAMPLE 6

PROBLEM SET B.1

Let f(x) � 3x and g(x) � (�12

�)x, and evaluate each of the following:

1. g(0) 2. f(0) 3. g(�1) 4. g(�4)5. f(�3) 6. f(�1) 7. f(2) � g(�2) 8. f (2) � g(�2)

Graph each of the following functions.

9. y � 4x 10. y � 2�x

11. y � 3�x 12. y � ��13

��x

13. y � 2x�1 14. y � 2x�3

15. y � ex 16. y � e�x

Graph each of the following functions on the same coordinate system for positivevalues of x only.

17. y � 2x, y � x2, y � 2x 18. y � 3x, y � x3, y � 3x

Graph the family of curves y � bx using the given values of the base b.

19. b � 2, 4, 6, 8 20. b � , , ,

21. Bouncing Ball Suppose the ball mentioned in the introduction to this section isdropped from a height of 6 feet above the ground. Find an exponential equationthat gives the height h the ball will attain during the nth bounce (Figure 7). Howhigh will it bounce on the fifth bounce?

1�8

1�6

1�4

1�2

After reading through the preceding section, respond in your own words andin complete sentences.

a. What is an exponential function?

b. In an exponential function, explain why the base b cannot equal 1. (Whatkind of function would you get if the base was equal to 1?)

c. Explain continuously compounded interest.

d. What characteristics do the graphs of y � 2x and y � ��12

��x

have in common?

GETTING READY FOR CLASS

22. Bouncing Ball A golf ball is manufactured so that if it is dropped from A feetabove the ground onto a hard surface, the maximum height of each bounce willbe �

12

� the height of the previous bounce. Find an exponential equation that givesthe height h the ball will attain during the nth bounce. If the ball is dropped from10 feet above the ground onto a hard surface, how high will it bounce on the 8thbounce?

23. Exponential Decay The half-life of iodine-131 is 8 days. If a patient is admin-istered a 1,400-microgram dose of iodine-131, how much iodine-131 will be inthe patient’s system after 8 days and after 11 days? (See Example 2.)

24. Exponential Growth Automobiles built before 1993 use Freon in their airconditioners. The federal government now prohibits the manufacture of Freon.Because the supply of Freon is decreasing, the price per pound is increasingexponentially. Current estimates put the formula for the price per pound of Freonat p(t) � 1.89(1.25)t, where t is the number of years since 1990. Find the priceof Freon in 2000 and 2005.

25. Compound Interest Suppose you deposit $1,200 in an account with an annualinterest rate of 6% compounded quarterly.a. Find an equation that gives the amount of money in the account after t years.b. Find the amount of money in the account after 8 years.c. If the interest were compounded continuously, how much money would the

account contain after 8 years?

26. Compound Interest Suppose you deposit $500 in an account with an annualinterest rate of 8% compounded monthly.a. Find an equation that gives the amount of money in the account after t years.b. Find the amount of money in the account after 5 years.c. If the interest were compounded continuously, how much money would the

account contain after 5 years?

27. Compound Interest If $5,000 is placed in an account with an annual interestrate of 12% compounded four times a year, how much money will be in theaccount 10 years later?

28. Compound Interest If $200 is placed in an account with an annual interest rateof 8% compounded twice a year, how much money will be in the account 10 years later?

29. Bacteria Growth Suppose it takes 1 day for a certain strain of bacteria toreproduce by dividing in half. If there are 100 bacteria present to begin with,then the total number present after x days will be f(x) � 100 � 2x. Find the totalnumber present after 1 day, 2 days, 3 days, and 4 days.


Bounce1

Bounce2

Bounce3

Bounce4

h

h•23

23

h23

Figure 7

30. Bacteria Growth Suppose it takes 12 hours for a certain strain of bacteria to re-produce by dividing in half. If there are 50 bacteria present to begin with, thenthe total number present after x days will be f(x) � 50 � 4x. Find the total num-ber present after 1 day, 2 days, and 3 days.

31. Health Care In 1990, $699 billion were spent on health care expenditures. Theamount of money, E, in billions spent on health care expenditures can be esti-mated using the function E(t) � 78.16(1.11)t, where t is time in years since1970 (U.S. Census Bureau).a. How close was the estimate determined by the function to the actual amount

of money spent on health care expenditures in 1990?b. What are the expected health care expenditures in 2005, 2006, and 2007?

Round to the nearest billion.

32. Exponential Growth The cost of a can of Coca-Cola on January 1, 1960, was10 cents. The following function gives the cost of a can of Coca-Cola t yearsafter that.

C(t) � 0.10e0.0576t

a. Use the function to fill in Table 2. (Round to the nearest cent.)


TABLE 2

Years Since 1960 Costt C(t)

0 $0.10

15

40

50

90

b. Use the table to find the cost of a can of Coca-Cola at the beginning of theyear 2000.

33. Value of a Painting A painting is purchased as an investment for $150. If thepainting’s value doubles every 3 years, then its value is given by the function

V(t) � 150 � 2t/3 for t � 0

where t is the number of years since it was purchased, and V(t) is its value (indollars) at that time. Graph this function.

34. Value of a Painting A painting is purchased as an investment for $125. If thepainting’s value doubles every 5 years, then its value is given by the function

V(t) � 125 � 2t/5 for t � 0

where t is the number of years since it was purchased, and V(t) is its value (indollars) at that time. Graph this function.

35. Value of a Crane The function

V(t) � 450,000(1 � 0.30)t

where V is value and t is time in years, can be used to find the value of a cranefor the first 6 years of use.a. What is the value of the crane after 3 years and 6 months?b. State the domain of this function.c. Sketch the graph of this function.

“Coca-Cola” is a registered trademarkof The Coca-Cola Company. Used withthe express permission of The Coca-Cola Company.

36. Value of a Printing Press The function V(t) � 375,000(1 � 0.25)t, where V isvalue and t is time in years, can be used to find the value of a printing press dur-ing the first 7 years of use.a. What is the value of the printing press after

4 years and 9 months?b. State the domain of this function.c. Sketch the graph of this function.

EXTENDING THE CONCEPTS

37. Drag Racing A dragster is equipped with a computer. The table gives the speedof the dragster every second during one race at the 1993 Winternationals. Fig-ure 8 is a line graph constructed from the data in Table 3.


0 1 2 3 4 5 60

50

100

150

200

250

Time (sec)

Spee

d (m

i/hr

)

Figure 8

TABLE 3 Speed of aDragster

Elapsed Time Speed(sec) (mi/hr)

0 0.0

1 72.7

2 129.9

3 162.8

4 192.2

5 212.4

6 228.1

The graph of the following function contains the first point and the last pointshown in Figure 8; that is, both (0, 0) and (6, 228.1) satisfy the function. Graphthe function to see how close it comes to the other points in Figure 8.

s(t) � 250(1 � 1.5�t)

38. The graphs of two exponential functions are given in Figures 9 and 10. Use thegraphs to find the following:a. f (0) b. f (�1) c. f (1) d. g(0) e. g(1) f. g(�1)

–3 –2 –1 1 2 3x

y

(0, 1)

(1, 3)

y = f(x)

2

3

4

5

6

7

1(–1, )13

Figure 9

–3 –2 –1 1 2 3

1

2

3

4

5

6

7

x

y

y = g(x)

(0, 1)(–1, 2)

(–2, 4)

(1, )12

Figure 10


SECTION B.2 LOGARITHMS ARE EXPONENTS

In January 1999, ABC News reported that an earthquake had occurred in Colombia,causing massive destruction. They reported the strength of the quake by indicatingthat it measured 6.0 on the Richter scale. For comparison, Table 1 gives the Richtermagnitude of a number of other earthquakes.

TABLE 1 Earthquakes

RichterYear Earthquake Magnitude

1971 Los Angeles 6.6

1985 Mexico City 8.1

1989 San Francisco 7.1

1992 Kobe, Japan 7.2

1994 Northridge 6.6

1999 Armenia, Colombia 6.0

Although the sizes of the numbers in the table do not seem to be very different, theintensity of the earthquakes they measure can be very different. For example, the1989 San Francisco earthquake was more than 10 times stronger than the 1999 earth-quake in Colombia. The reason behind this is that the Richter scale is a logarithmicscale. In this section, we start our work with logarithms, which will give you anunderstanding of the Richter scale. Let’s begin.

As you know from your work in the previous section, equations of the form

y � bx b � 0, b � 1

are called exponential functions. Every exponential function is a one-to-one func-tion; therefore, it will have an inverse that is also a function. Because the equation ofthe inverse of a function can be obtained by exchanging x and y in the equation of theoriginal function, the inverse of an exponential function must have the form

x � by b � 0, b � 1

Now, this last equation is actually the equation of a logarithmic function, as the fol-lowing definition indicates:

DEFINITION

The equation y � logb x is read “y is the logarithm to the base b of x” and isequivalent to the equation

x � by b � 0, b � 1

In words: y is the exponent to which b is raised in order to get x.

Notation When an equation is in the form x � by, it is said to be in exponentialform. On the other hand, if an equation is in the form y � logb x, it is said to be inlogarithmic form.

Table 2 shows some equivalent statements written in both forms.

475Section B.2 Logarithms Are Exponents

TABLE 2

Exponential Form Logarithmic Form

8 � 23 ⇐⇒ log2 8 � 3

25 � 52 ⇐⇒ log5 25 � 2

0.1 � 10�1 ⇐⇒ log10 0.1 � �1

� 2�3 ⇐⇒ log2 � �3

r � zs ⇐⇒ logz r � s

1�8

1�8

Solve log3 x � �2.

SOLUTION In exponential form the equation looks like this:

x � 3�2

or x �

The solution is .

Solve logx 4 � 3.

SOLUTION Again, we use the definition of logarithms to write the equation inexponential form:

4 � x3

Taking the cube root of both sides, we have

�3

4� � �3

x3�x � �

34�

The solution is �3

4�.

Solve log8 4 � x.

SOLUTION We write the equation again in exponential form:

4 � 8x

Because both 4 and 8 can be written as powers of 2, we write them in terms of pow-ers of 2:

22 � (23)x

22 � 23x

The only way the left and right sides of this last line can be equal is if the exponentsare equal—that is, if

2 � 3x

or x �2�3

1�9

1�9

EXAMPLE 1

EXAMPLE 2

EXAMPLE 3

The solution is �23

�. We check as follows:

log8 4 � ⇐⇒ 4 � 82/3

4 � (�3

8�)2

4 � 22

4 � 4

The solution checks when used in the original equation.

Graphing Logarithmic Functions

Graphing logarithmic functions can be done using the graphs of exponential func-tions and the fact that the graphs of inverse functions have symmetry about the liney � x. Here’s an example to illustrate.

Graph the equation y � log2 x.

SOLUTION The equation y � log2 x is, by definition, equivalent to the exponen-tial equation

x � 2y

which is the equation of the inverse of the function

y � 2x

We simply reflect the graph of y � 2x about the line y � x to get the graph of x � 2y,which is also the graph of y � log2 x. (See Figure 1.)

2�3


EXAMPLE 4

–5 –4 –3 –2 –1 1 2 3 4 5

–5–4–3–2–1

12

45

x

y

3y = x

y = 2x

y = log2xor

x = 2y

Figure 1

It is apparent from the graph that y � log2 x is a function because no vertical linewill cross its graph in more than one place. The same is true for all logarithmic equa-tions of the form y � logb x, where b is a positive number other than 1. Note also thatthe graph of y � logb x will always appear to the right of the y-axis, meaning that xwill always be positive in the equation y � logb x.

GRAPHING LOGARITHMIC FUNCTIONS

As demonstrated in Example 4, we can graph the logarithmic function y � log2 x asthe inverse of the exponential function y � 2x. Your graphing calculator most likelyhas a command to do this. First, define the exponential function as Y1 � 2x. Tosee the line of symmetry, define a second function Y2 � x. Set the window variablesso that

�6 x 6; �6 y 6

and use your zoom-square command to graph both functions. Your graph should looksimilar to the one shown in Figure 2. Now use the appropriate command to graph theinverse of the exponential function defined as Y1 (Figure 3).


USING TECHNOLOGY

Figure 2

DrawInv Y1

Figure 3

Two Special Identities

If b is a positive real number other than 1, then each of the following is a consequenceof the definition of a logarithm:

(1) b logb x � x and (2) logb bx � x

The justifications for these identities are similar. Let’s consider only the first one.Consider the equation

y � logb x

By definition, it is equivalent to

x � by

Substituting logb x for y in the last line gives us

x � b logb x

The next examples in this section show how these two special properties can beused to simplify expressions involving logarithms.

Simplify 2log23.

SOLUTION Using the first property, 2log23 � 3.

Simplify log10 10,000.

SOLUTION 10,000 can be written as 104:

log10 10,000 � log10 104

� 4

Simplify logb b (b � 0, b � 1).

SOLUTION Because b1 � b, we have

logb b � logb b1

� 1

Simplify logb 1 (b � 0, b � 1).

SOLUTION Because 1 � b0, we have

logb 1 � logb b0

� 0

Simplify log4 (log5 5).

SOLUTION Because log5 5 � 1,

log4 (log5 5) � log4 1

� 0

Application

As we mentioned in the introduction to this section, one application of logarithms isin measuring the magnitude of an earthquake. If an earthquake has a shock waveT times greater than the smallest shock wave that can be measured on a seismograph(Figure 4), then the magnitude M of the earthquake, as measured on the Richter scale,is given by the formula

M � log10 T

(When we talk about the size of a shock wave, we are talking about its amplitude. Theamplitude of a wave is half the difference between its highest point and its lowest point.)

To illustrate the discussion, an earthquake that produces a shock wave that is10,000 times greater than the smallest shock wave measurable on a seismograph willhave a magnitude M on the Richter scale of

M � log10 10,000 � 4


EXAMPLE 6

EXAMPLE 7

EXAMPLE 8

EXAMPLE 9

EXAMPLE 5

If an earthquake has a magnitude of M � 5 on the Richterscale, what can you say about the size of its shock wave?

SOLUTION To answer this question, we put M � 5 into the formula M � log10 Tto obtain

5 � log10 T

Writing this equation in exponential form, we have

T � 105 � 100,000

We can say that an earthquake that measures 5 on the Richter scale has a shock wave 100,000 times greater than the smallest shock wave measurable on a seis-mograph.

From Example 10 and the discussion that preceded it, we find that an earthquakeof magnitude 5 has a shock wave that is 10 times greater than an earthquake of mag-nitude 4 because 100,000 is 10 times 10,000.


14:00:00

Sat Apr 25 1992GMT

+90s +180s +270s +360s +450s +540s +630s +720s +810s +900s

16:00:00

18:00:00

20:00:00

22:00:00

00:00:00

BK

S L

HZ

02:00:00

04:00:00

06:00:00

08:00:00

10:00:00

12:00:00

Figure 4

EXAMPLE 10


a. What is a logarithm?

b. What is the relationship between y � 2x and y � log2 x? How are theirgraphs related?

c. Will the graph of y � logb x ever appear in the second or third quadrants?Explain why or why not.

d. Explain why log2 0 � x has no solution for x.



PROBLEM SET B.2

Write each of the following equations in logarithmic form.

1. 24 � 16 2. 32 � 9 3. 125 � 53 4. 16 � 42

5. 0.01 � 10�2 6. 0.001 � 10�3 7. 2�5 � 8. 4�2 �

9. � ��3

� 8 10. � ��2

� 9 11. 27 � 33 12. 81 � 34

Write each of the following equations in exponential form.

13. log10 100 � 2 14. log2 8 � 3 15. log2 64 � 6 16. log2 32 � 517. log8 1 � 0 18. log9 9 � 1 19. log6 36 � 2 20. log7 49 � 2

21. log10 0.001 � �3 22. log10 0.0001 � �4

23. log5 � �2 24. log3 � �4

Solve each of the following equations for x.

25. log3 x � 2 26. log4 x � 3 27. log5 x � �3 28. log2 x � �429. log2 16 � x 30. log3 27 � x 31. log8 2 � x 32. log25 5 � x33. logx 4 � 2 34. logx 16 � 4 35. logx 5 � 3 36. logx 8 � 2

Sketch the graph of each of the following logarithmic equations.

37. y � log3 x 38. y � log1/2 x 39. y � log1/3 x 40. y � log4 x

Use your graphing calculator to graph each exponential function with the zoom-square command. Then use the appropriate command to graph the inverse function,and write its equation in logarithmic form.

41. y � 5x 42. y � � �x

43. y � 10x 44. y � ex

Simplify each of the following.

45. log2 16 46. log3 9 47. log25 125 48. log9 2749. log10 1,000 50. log10 10,000 51. log3 3 52. log4 453. log5 1 54. log10 1 55. log3 (log6 6) 56. log5 (log3 3)

MeasuringAcidity In chemistry, the pH of a solution is defined in terms of logarithmsas pH � �log10 [H�] where [H�] is the hydrogen ion concentration in the solution.Anacid solution has a pH below 7, and a basic solution has a pH higher than 7.

57. In distilled water, the hydrogen ion concentration is [H�] � 10�7. What is the pH?58. Find the pH of a bottle of vinegar if the hydrogen ion concentration is

[H�] � 10�3.59. A hair conditioner has a pH of 6. Find the hydrogen ion concentration, [H�], in

the conditioner.60. If a glass of orange juice has a pH of 4, what is the hydrogen ion concentration,

[H�], in the orange juice?

61. Magnitude of an Earthquake Find the magnitude M of an earthquake with ashock wave that measures T � 100 on a seismograph.

62. Magnitude of an Earthquake Find the magnitude M of an earthquake with ashock wave that measures T � 100,000 on a seismograph.

63. Shock Wave If an earthquake has a magnitude of 8 on the Richter scale, howmany times greater is its shock wave than the smallest shock wave measurableon a seismograph?

1�5

1�81

1�25

1�3

1�2

1�16

1�32

64. Shock Wave If an earthquake has a magnitude of 6 on the Richter scale, howmany times greater is its shock wave than the smallest shock wave measurableon a seismograph?


65. The graph of the exponential function y � f(x) � bx is shown in Figure 5. Usethe graph to complete parts a through d.


–5 –4 –3 –2 1 2 3 4 5

–64

–48

–32

–16

16

32

48

64

x

y

(1, 8)

Figure 5

a. Fill in the table. b. Fill in the table.

x f (x)

�1

0

1

2

x f�1(x)

�1

0

1

2

c. Find the equation for f(x). d. Find the equation for f�1(x).

66. The graph of the exponential function y � f(x) � bx is shown in Figure 6. Usethe graph to complete parts a through d.

–5 –4 –3 –2 –1 1 2 3 4 5

–35–30–25–20–15–10

101520253035

x

y

(–2, 25)

Figure 6

c. Find the equation for f(x). d. Find the equation for f�1(x).


a. Fill in the table. b. Fill in the table.

x f (x)

�1

0

1

2

x f�1(x)

�1

0

1

2

SECTION B.3 PROPERTIES OF LOGARITHMS

If we search for the word decibel in Microsoft Bookshelf 98, we find the followingdefinition:

A unit used to express relative difference in power or intensity, usually betweentwo acoustic or electric signals, equal to ten times the common logarithm of theratio of the two levels.

Figure 1 shows the decibel rating associated with a number of common sounds.

Loud conversation

Noisy office

Light traffic

Light whisper

Normal conversation

Quiet conversation

Normal traffic, quiet train

Rock music, subway

Heavy traffic, thunder

Jet plane at takeoff

11010090807060

Decibels

50403020100

Figure 1

The precise definition for a decibel is

D � 10 log10 � �where I is the intensity of the sound being measured, and I0 is the intensity of the leastaudible sound. (Sound intensity is related to the amplitude of the sound wave thatmodels the sound and is given in units of watts per meter2.) In this section, we willsee that the preceding formula can also be written as

D � 10(log10 I � log10 I0)

The rules we use to rewrite expressions containing logarithms are called the proper-ties of logarithms. There are three of them.

I�I0

For the following three properties, x, y, and b are all positive real numbers,b 1, and r is any real number.

483Section B.3 Properties of Logarithms

PROPERTY 1

logb (xy) � logb x � logb y

In words: The logarithm of a product is the sum of the logarithms.

PROPERTY 2

logb � � � logb x � logb y

In words: The logarithm of a quotient is the difference of the logarithms.

x�y

PROPERTY 3

logb xr � r logb x

In words: The logarithm of a number raised to a power is the product of thepower and the logarithm of the number.

PROOF OF PROPERTY 1

To prove property 1, we simply apply the first identity for logarithms given in the pre-ceding section:

b logb xy � xy � (b logb x)(b logb y) � b logb x�logb y

The first and last expressions are equal and the bases are the same, so the exponentslogb xy and logb x � logb y must be equal. Therefore,

logb xy � logb x � logb y

The proofs of properties 2 and 3 proceed in much the same manner, so we willomit them here. The examples that follow show how the three properties can be used.

Expand log5 using the properties of logarithms.

SOLUTION Applying property 2, we can write the quotient of 3xy and z in termsof a difference:

log5 � log5 3xy � log5 z

Applying property 1 to the product 3xy, we write it in terms of addition:

log5 � log5 3 � log5 x � log5 y � log5 z3xy�

z

3xy�

z

3xy�

zEXAMPLE 1

Expand, using the properties of logarithms:

log2

SOLUTION We write �y� as y1/2 and apply the properties:

log2 � log2 �y� � y1/2

� log2 x4 � log2 (y1/2 � z3) Property 2

� log2 x4 � (log2 y1/2 � log2 z3) Property 1

� log2 x4 � log2 y1/2 � log2 z3 Remove parentheses

� 4 log2 x � log2 y � 3 log2 z Property 3

We can also use the three properties to write an expression in expanded form asjust one logarithm.

Write as a single logarithm:

2 log10 a � 3 log10 b � log10 c

SOLUTION We begin by applying property 3:

2 log10 a � 3 log10 b � log10 c

� log10 a2 � log10 b3 � log10 c1/3 Property 3

� log10 (a2 � b3) � log10 c1/3 Property 1

� log10 Property 2

� log10 c1/3 � �3

c�

The properties of logarithms along with the definition of logarithms are useful insolving equations that involve logarithms.

Solve log2 (x � 2) � log2 x � 3.

SOLUTION Applying property 1 to the left side of the equation allows us towrite it as a single logarithm:

log2 (x � 2) � log2 x � 3

log2 [(x � 2)(x)] � 3

The last line can be written in exponential form using the definition of logarithms:

(x � 2)(x) � 23

Solve as usual:

x2 � 2x � 8

x2 � 2x � 8 � 0

(x � 4)(x � 2) � 0

x � 4 � 0 or x � 2 � 0

x � �4 or x � 2

a2b3

��

3c�

a2b3

�c1/3

1�3

1�3

1�2

x4

�y1/2z3

x4

��y� � z3

x4

��y� � z3


EXAMPLE 2

EXAMPLE 3

EXAMPLE 4

In the previous section we noted the fact that x in the expression y � logb x cannot bea negative number. Because substitution of x � �4 into the original equation gives

log2 (�2) � log2 (�4) � 3

which contains logarithms of negative numbers, we cannot use �4 as a solution. Thesolution set is {2}.

485Section B.3 Properties of Logarithms


a. Explain why the following statement is false: “The logarithm of a productis the product of the logarithms.”

b. Explain why the following statement is false: “The logarithm of a quotientis the quotient of the logarithms.”

c. Explain the difference between logb m � logb n and logb (m � n). Are theyequivalent?

d. Explain the difference between logb (mn) and (logb m)(logb n). Are theyequivalent?


Use the three properties of logarithms given in this section to expand each expressionas much as possible.

1. log3 4x 2. log2 5x 3. log6 4. log3

5. log2 y5 6. log7 y3 7. log9 �3

z� 8. log8 �z�9. log6 x2y4 10. log10 x2y4 11. log5 �x� � y4 12. log8 �3 xy6�

13. logb 14. logb 15. log10 16. log10

17. log10 18. log10 19. logb �3 � 20. logb �4 �Write each expression as a single logarithm.

21. logb x � logb z 22. logb x � logb z23. 2 log3 x � 3 log3 y 24. 4 log2 x � 5 log2 y

25. log10 x � log10 y 26. log10 x � log10 y

27. 3 log2 x � log2 y � log2 z 28. 2 log3 x � 3 log3 y � log3 z

29. log2 x � 3 log2 y � 4 log2 z 30. 3 log10 x � log10 y � log10 z

31. log10 x � log10 y � log10 z 32. 3 log10 x � log10 y � 5 log10 z4�3

4�5

3�4

3�2

1�2

1�2

1�4

1�3

1�3

1�2

x4y3

�z5

x2y�z4

x4�3

y��

�z�

x3�y��

z4

5�4y

4�xy

3x�y

xy�z

x�5

5�x

PROBLEM SET B.3

Solve each of the following equations.

33. log2 x � log2 3 � 1 34. log3 x � log3 3 � 135. log3 x � log3 2 � 2 36. log3 x � log3 2 � 237. log3 x � log3 (x � 2) � 1 38. log6 x � log6 (x � 1) � 139. log3 (x � 3) � log3 (x � 1) � 1 40. log4 (x � 2) � log4 (x � 1) � 141. log2 x � log2 (x � 2) � 3 42. log4 x � log4 (x � 6) � 2

43. log8 x � log8 (x � 3) � 44. log27 x � log27 (x � 8) �

45. log5 �x� � log5 �6x � 5� � 1 46. log2 �x� � log2 �6x � 5� � 1

47. Food Processing The formula M � 0.21(log10 a � log10 b) is used in the foodprocessing industry to find the number of minutes M of heat processing a certainfood should undergo at 250°F to reduce the probability of survival of Clostrid-ium botulinum spores. The letter a represents the number of spores per can be-fore heating, and b represents the number of spores per can after heating. Find Mif a � 1 and b � 10�12. Then find M using the same values for a and b in theformula

M � 0.21 log10

48. Acoustic Powers The formula N � log10 is used in radio electronics to find

the ratio of the acoustic powers of two electric circuits in terms of their electricpowers. Find N if P1 is 100 and P2 is 1. Then use the same two values of P1 andP2 to find N in the formula N � log10 P1 � log10 P2.

49. Henderson–Hasselbalch Formula Doctors use the Henderson–Hasselbalchformula to calculate the pH of a person’s blood. pH is a measure of the acidityand/or the alkalinity of a solution. This formula is represented as

pH � 6.1 � log10 � �where x is the base concentration and y is the acidic concentration. Rewrite theHenderson–Hasselbalch formula so that the logarithm of a quotient is not in-volved.

50. Henderson–Hasselbalch Formula Refer to the information in the precedingproblem about the Henderson–Hasselbalch formula. If most people have a bloodpH of 7.4, use the Henderson–Hasselbalch formula to find the ratio of x/y for anaverage person.

51. Decibel Formula Use the properties of logarithms to rewrite the decibel for-

mula D � 10 log10 � � so that the logarithm of a quotient is not involved.

52. Decibel Formula In the decibel formula D � 10 log10 � �, the threshold ofhearing, I0, is

I0 � 10�12 watts/meter2

Substitute 10�12 for I0 in the decibel formula, and then show that it simplifies to

D � 10(log10 I � 12)

I�I0

I�I0

x�y

P1�P2

a�b

2�3

2�3


Common Logarithms

Two kinds of logarithms occur more frequently than other logarithms. Logarithmswith a base of 10 are very common because our number system is a base-10 numbersystem. For this reason, we call base-10 logarithms common logarithms.

487Section B.4 Common Logarithms and Natural Logarithms

SECTION B.4 COMMON LOGARITHMS AND NATURAL LOGARITHMS

Acid rain was first discovered in the 1960s by Gene Likens and his research team,who studied the damage caused by acid rain to Hubbard Brook in New Hampshire.Acid rain is rain with a pH of 5.6 and below. As you will see as you work your waythrough this section, pH is defined in terms of common logarithms—one of the top-ics we present in this section. So, when you are finished with this section, you willhave a more detailed knowledge of pH and acid rain.

DEFINITION

A common logarithm is a logarithm with a base of 10. Because common loga-rithms are used so frequently, it is customary, in order to save time, to omitnotating the base; that is,

log10 x � log x

When the base is not shown, it is assumed to be 10.

Common logarithms of powers of 10 are simple to evaluate. We need only recognize that log 10 � log10 10 � 1 and apply the third property of logarithms:logb xr � r logb x.

log 1,000 � log 103 � 3 log 10 � 3(1) � 3

log 100 � log 102 � 2 log 10 � 2(1) � 2

log 10 � log 101 � 1 log 10 � 1(1) � 1

log 1 � log 100 � 0 log 10 � 0(1) � 0

log 0.1 � log 10�1 � �1 log 10 � �1(1) � �1

log 0.01 � log 10�2 � �2 log 10 � �2(1) � �2

log 0.001 � log 10�3 � �3 log 10 � �3(1) � �3

To find common logarithms of numbers that are not powers of 10, we use a cal-culator with a key. log

SO2, NO2

HNO3

H2SO4

(acids)

Hundredsof miles

Acid rain

Check the following logarithms to be sure you know how to use your calculator.(These answers have been rounded to the nearest ten-thousandth.)

log 7.02 � 0.8463

log 1.39 � 0.1430

log 6.00 � 0.7782

log 9.99 � 0.9996

Use a calculator to find log 2,760.

SOLUTION log 2,760 � 3.4409

To work this problem on a scientific calculator, we simply enter the number 2,760and press the key labeled . On a graphing calculator we press the key first,then 2,760.

The 3 in the answer is called the characteristic, and the decimal part of the log-arithm is called the mantissa.

Find log 0.0391.

SOLUTION log 0.0391 � �1.4078

Find log 0.00523.

SOLUTION log 0.00523 � �2.2815

Find x if log x � 3.8774.

SOLUTION We are looking for the number whose logarithm is 3.8774. On ascientific calculator, we enter 3.8774 and press the key labeled . On a graphingcalculator we press first, then 3.8774. The result is 7,540 to four significant dig-its. Here’s why:

If log x � 3.8774

then x � 103.8774

� 7,540

The number 7,540 is called the antilogarithm or just antilog of 3.8774. That is, 7,540is the number whose logarithm is 3.8774.

Find x if log x � �2.4179.

SOLUTION Using the key, the result is 0.00382.

If log x � �2.4179

then x � 10�2.4179

� 0.00382

The antilog of �2.4179 is 0.00382; that is, the logarithm of 0.00382 is �2.4179.

Applications

Previously, we found that the magnitude M of an earthquake that produces a shock waveT times larger than the smallest shock wave that can be measured on a seismograph is

10 x

10 x10 x

loglog


EXAMPLE 1

EXAMPLE 2

EXAMPLE 3

EXAMPLE 4

EXAMPLE 5

given by the formulaM � log10 T

We can rewrite this formula using our shorthand notation for common logarithms as

M � log T

The San Francisco earthquake of 1906 is estimatedto have measured 8.3 on the Richter scale. The San Fernando earthquake of 1971measured 6.6 on the Richter scale. Find T for each earthquake, and then give someindication of how much stronger the 1906 earthquake was than the 1971 earthquake.

SOLUTION For the 1906 earthquake:

If log T � 8.3, then T � 108.3 � 2.00 � 108

For the 1971 earthquake:

If log T � 6.6, then T � 106.6 � 3.98 � 106

Dividing the two values of T and rounding our answer to the nearest whole number,we have

� 50

The shock wave for the 1906 earthquake was approximately 50 times stronger thanthe shock wave for the 1971 earthquake.

In chemistry, the pH of a solution is the measure of the acidity of the solution.The definition for pH involves common logarithms. Here it is:

pH � �log [H�]

where [H�] is the hydrogen ion concentration. The range for pH is from 0 to 14 (Fig-ure 1). Pure water, a neutral solution, has a pH of 7. An acidic solution, such as vinegar,will have a pH less than 7, and an alkaline solution, such as ammonia, has a pH above 7.

2.00 � 108

��3.98 � 106


EXAMPLE 6

Bat

tery

aci

d

Mos

t aci

dic

rain

fall

reco

rded

in U

.S.

Lem

on ju

ice

Vin

egar

App

le ju

ice

Ave

rage

pH

of

rain

fall,

Tor

onto

, Feb

. 197

9To

mat

o ju

ice

Ave

rage

pH

of

Kill

arne

y L

akes

, 197

1M

ean

pH o

f Adi

rond

ack

Lak

es, 1

975

"Cle

an"

rain

Upp

er li

mit

at w

hich

som

e fi

sh a

ffec

ted

Mea

n pH

of A

diro

ndac

k L

akes

, 193

0M

ilk

Blo

od

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ntar

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awat

er, b

akin

g so

da

Milk

of

mag

nesi

a

Am

mon

ia

Lye

ACID RAIN

Increasingly acidicIncreasingly alkaline

1 0234567891011121314

NE

UT

RA

LN

EU

TR

AL

THE ACID SCALE

Figure 1

8.3

Normal rainwater has a pH of 5.6. What is the hydrogenion concentration in normal rainwater?

SOLUTION Substituting 5.6 for pH in the formula pH � �log [H�], we have

5.6 � �log [H�] Substitution

log [H�] � �5.6 Isolate the logarithm

[H�] � 10�5.6 Write in exponential form

� 2.5 � 10�6 moles per liter Answer in scientific notation

The hydrogen ion concentration in a sample of acid rain known to kill fish is 3.2 � 10�5 moles per liter. Find the pH of this acid rain to thenearest tenth.

SOLUTION Substituting 3.2 � 10�5 for [H�] in the formula pH � �log [H�],we have

pH � �log [3.2 � 10�5] Substitution

pH � �(�4.5) Evaluate the logarithm

� 4.5 Simplify

Natural Logarithms


EXAMPLE 7

EXAMPLE 8

DEFINITION

A natural logarithm is a logarithm with a base of e. The natural logarithm of xis denoted by ln x; that is,

ln x � loge x

The postage stamp shown in Figure 2 contains one of the two special identities wementioned previously in this chapter, but stated in terms of natural logarithms.

Figure 2

We can assume that all our properties of exponents and logarithms hold forexpressions with a base of e because e is a real number. Here are some examplesintended to make you more familiar with the number e and natural logarithms.

Simplify each of the following expressions.

a. e0 � 1

b. e1 � e

c. ln e � 1 In exponential form, e1 � e

d. ln 1 � 0 In exponential form, e0 � 1

e. ln e3 � 3

f. ln e�4 � �4

g. ln et � t

Use the properties of logarithms to expand the expres-sion ln Ae5t.

SOLUTION The properties of logarithms hold for natural logarithms, so we have

ln Ae5t � ln A � ln e5t

� ln A � 5t ln e

� ln A � 5t Because ln e � 1

If ln 2 � 0.6931 and ln 3 � 1.0986, find

a. ln 6 b. ln 0.5 c. ln 8

SOLUTION

a. Because 6 � 2 � 3, using Property 1 for logarithms we have

ln 6 � ln (2 � 3)

� ln 2 � ln 3

� 0.6931 � 1.0986

� 1.7917

b. Writing 0.5 as �12

� and applying property 2 for logarithms gives us

ln 0.5 � ln

� ln 1 � ln 2

� 0 � 0.6931

� �0.6931

c. Writing 8 as 23 and applying property 3 for logarithms, we have

ln 8 � ln 23

� 3 ln 2

� 3(0.6931)

� 2.0793

1�2


EXAMPLE 9

EXAMPLE 10

EXAMPLE 11

Find the following logarithms.

1. log 378 2. log 37.8 3. log 3,780 4. log 0.42605. log 0.0378 6. log 37,800 7. log 600 8. log 10,2009. log 0.00971 10. log 0.0314 11. log 0.00052 12. log 0.399

Find x in the following equations.

13. log x � 2.8802 14. log x � 4.8802 15. log x � �2.119816. log x � �3.1198 17. log x � �5.3497 18. log x � �1.5670

Find x without using a calculator.

19. log x � 10 20. log x � �1 21. log x � �1022. log x � 1 23. log x � 20 24. log x � �2025. log x � �2 26. log x � 4 27. log x � log2 828. log x � log3 9

Simplify each of the following expressions.

29. ln e 30. ln 1 31. ln e�3 32. ln ex

Use the properties of logarithms to expand each of the following expressions.

33. ln 10e3t 34. ln 10e4t 35. ln Ae�2t 36. ln Ae�3t

Find a decimal approximation to each of the following natural logarithms.

37. ln 345 38. ln 3,450 39. ln 0.345 40. ln 0.034541. ln 10 42. ln 100 43. ln 45,000 44. ln 450,000

If ln 2 � 0.6931, ln 3 � 1.0986, and ln 5 � 1.6094, find each of the following.

45. ln 15 46. ln 10 47. ln 48. ln

49. ln 9 50. ln 25 51. ln 16 52. ln 81

Measuring Acidity Previously we indicated that the pH of a solution is defined interms of logarithms as

pH � �log [H�]

where [H�] is the hydrogen ion concentration in that solution.

53. Find the pH of orange juice if the hydrogen ion concentration in the juice is[H�] � 6.50 � 10�4.

1�5

1�3



a. What is a common logarithm?

b. What is a natural logarithm?

c. Is e a rational number? Explain.

d. Find ln e, and explain how you arrived at your answer.


PROBLEM SET B.4

54. Find the pH of milk if the hydrogen ion concentration in milk is [H�] � 1.88 � 10�6.

55. Find the hydrogen ion concentration in a glass of wine if the pH is 4.75.56. Find the hydrogen ion concentration in a bottle of vinegar if the pH is 5.75.

The Richter Scale Find the relative size T of the shock wave of earthquakes with thefollowing magnitudes, as measured on the Richter scale.

57. 5.5 58. 6.6 59. 8.3 60. 8.7

61. Shock Wave How much larger is the shock wave of an earthquake that mea-sures 6.5 on the Richter scale than one that measures 5.5 on the same scale?

62. Shock Wave How much larger is the shock wave of an earthquake that mea-sures 8.5 on the Richter scale than one that measures 5.5 on the same scale?

63. Earthquake Table 1 gives a partial listing of earthquakes that were recorded inCanada in 2000. Complete the table by computing the magnitude on the Richterscale, M, or the number of times the associated shock wave is larger than thesmallest measurable shock wave, T.


TABLE 1

Location Date Magnitude, M Shock Wave, T

Moresby Island January 23 4.0

Vancouver Island April 30 1.99 � 105

Quebec City June 29 3.2

Mould Bay November 13 5.2

St. Lawrence December 14 5.01 � 103

Source: National Resources Canada, National Earthquake Hazards Program.

64. Earthquake On January 6, 2001, an earthquake with a magnitude of 7.7 on theRichter scale hit southern India (National Earthquake Information Center). Bywhat factor was this earthquake’s shock wave greater than the smallest measur-able shock wave?

Depreciation The annual rate of depreciation r on a car that is purchased for P dol-lars and is worth W dollars t years later can be found from the formula

log (1 � r) � log �W

P�

65. Find the annual rate of depreciation on a car that is purchased for $9,000 andsold 5 years later for $4,500.

66. Find the annual rate of depreciation on a car that is purchased for $9,000 andsold 4 years later for $3,000.

Two cars depreciate in value according to the following depreciation tables. In eachcase, find the annual rate of depreciation.

67. 68.

1�t

Age in Value inYears Dollars

new 7,550

5 5,750

Age in Value inYears Dollars

new 7,550

3 5,750

69. Getting Close to e Use a calculator to complete the following table.


x (1 � x)1/x

1

0.5

0.1

0.01

0.001

0.0001

0.00001

What number does the expression (1 � x)1/x seem to approach as x gets closerand closer to zero?

70. Getting Close to e Use a calculator to complete the following table.

x �1 � �1x

��x

1

10

50

100

500

1,000

10,000

1,000,000

What number does the expression �1 � �1

x��

xseem to approach as x gets larger

and larger?

SECTION B.5 EXPONENTIAL EQUATIONSAND CHANGE OF BASE

For items involved in exponential growth, the time it takes for a quantity to double iscalled the doubling time. For example, if you invest $5,000 in an account that pays5% annual interest, compounded quarterly, you may want to know how long it willtake for your money to double in value. You can find this doubling time if you cansolve the equation

10,000 � 5,000(1.0125)4t

As you will see as you progress through this section, logarithms are the key to solv-ing equations of this type.

Logarithms are very important in solving equations in which the variable ap-pears as an exponent. The equation

5x � 12

is an example of one such equation. Equations of this form are called exponentialequations. Because the quantities 5x and 12 are equal, so are their common loga-rithms. We begin our solution by taking the logarithm of both sides:

log 5x � log 12

We now apply property 3 for logarithms, log xr � r log x, to turn x from an ex-ponent into a coefficient:

x log 5 � log 12

Dividing both sides by log 5 gives us

x � �l

l

o

o

g

g

1

5

2�

If we want a decimal approximation to the solution, we can find log 12 and log 5 on a calculator and divide:

x � �1

0

.

.

0

6

7

9

9

9

2

0�

� 1.5439

The complete problem looks like this:

5x � 12

log 5x � log 12

x log 5 � log 12

x � �l

l

o

o

g

g

1

5

2�

� �1

0

.

.

0

6

7

9

9

9

2

0�

� 1.5439

Here is another example of solving an exponential equation using logarithms.

Solve 252x�1 � 15.

SOLUTION Taking the logarithm of both sides and then writing the exponent(2x � 1) as a coefficient, we proceed as follows:

252x�1 � 15

log 252x�1 � log 15 Take the log of both sides

(2x � 1) log 25 � log 15 Property 3

2x � 1 � �l

l

o

o

g

g

1

2

5

5� Divide by log 25

2x � �l

l

o

o

g

g

1

2

5

5� � 1 Add �1 to both sides

x � �1

2��ll

o

o

g

g

1

2

5

5� � 1� Multiply both sides by �

12

�

495Section B.5 Exponential Equations and Change of Base

EXAMPLE 1

Using a calculator, we can write a decimal approximation to the answer:

x � �1

2��11

.

.

1

3

7

9

6

7

1

9� � 1�

� �1

2�(0.8413 � 1)

� �1

2�(�0.1587)

� �0.0794

Recall from Section B.1 that if you invest P dollars in an account with an annualinterest rate r that is compounded n times a year, then t years later the amount ofmoney in that account will be

A � P�1 � �n

r��

nt

How long does it take for $5,000 to double if it isdeposited in an account that yields 5% interest compounded once a year?

SOLUTION Substituting P � 5,000, r � 0.05, n � 1, and A � 10,000 into ourformula, we have

10,000 � 5,000(1 � 0.05)t

10,000 � 5,000(1.05)t

2 � (1.05)t Divide by 5,000

This is an exponential equation. We solve by taking the logarithm of both sides:

log 2 � log (1.05)t

� t log 1.05

Dividing both sides by log 1.05, we have

t � �lo

l

g

og

1.

2

05�

� 14.2

It takes a little over 14 years for $5,000 to double if it earns 5% interest per year,compounded once a year.

There is a fourth property of logarithms we have not yet considered. This lastproperty allows us to change from one base to another and is therefore called thechange-of-base property.


EXAMPLE 2

PROPERTY 4 (CHANGE OF BASE)

If a and b are both positive numbers other than 1, and if x � 0, then

loga x � �l

l

o

o

g

g

b

b

a

x�

↑ ↑Base a Base b

The logarithm on the left side has a base of a, and both logarithms on the right sidehave a base of b. This allows us to change from base a to any other base b that is apositive number other than 1. Here is a proof of property 4 for logarithms.

PROOF

We begin by writing the identity

a loga x � x

Taking the logarithm base b of both sides and writing the exponent loga x as a coef-ficient, we have

logb a loga x � logb x

loga x logb a � logb x

Dividing both sides by logb a, we have the desired result:

�log

la

o

x

g

l

b

o

a

gb a� � �

l

l

o

o

g

g

b

b

a

x�

loga x � �l

l

o

o

g

g

b

b

a

x�

We can use this property to find logarithms we could not otherwise compute on ourcalculators—that is, logarithms with bases other than 10 or e. The next exampleillustrates the use of this property.

Find log8 24.

SOLUTION We do not have base-8 logarithms on our calculators, so wecan change this expression to an equivalent expression that contains only base-10logarithms:

log8 24 � �l

l

o

o

g

g

2

8

4� Property 4

Don’t be confused. We did not just drop the base, we changed to base 10. We couldhave written the last line like this:

log8 24 � �l

l

o

o

g

g1

1

0

0

2

8

4�

From our calculators, we write

log8 24 � �1

0

.

.

3

9

8

0

0

3

2

1�

� 1.5283

Application

Suppose that the population in a small city is 32,000 inthe beginning of 1994 and that the city council assumes that the population sizet years later can be estimated by the equation

P � 32,000e0.05t

Approximately when will the city have a population of 50,000?


EXAMPLE 3

EXAMPLE 4

SOLUTION We substitute 50,000 for P in the equation and solve for t:

50,000 � 32,000e0.05t

1.5625 � e0.05t Divide both sides by 32,000

To solve this equation for t, we can take the natural logarithm of each side:

ln 1.5625 � ln e0.05t

� 0.05t ln e Property 3 for logarithms

� 0.05t Because ln e � 1

t � �ln

0

1

.

.

0

5

5

625� Divide each side by 0.05

� 8.93 years

We can estimate that the population will reach 50,000 toward the end of 2002.


USING TECHNOLOGY

SOLVING EXPONENTIAL EQUATIONS

We can solve the equation 50,000 � 32,000e0.05t from Example 4 with a graphing cal-culator by defining the expression on each side of the equation as a function. Thesolution to the equation will be the x-value of the point where the two graphs intersect.

First define functions Y1 � 50000 and Y2 � 32000e (0.05x) as shown in Figure 1.Set your window variables so that

0 x 15; 0 y 70,000, scale � 10,000

Graph both functions, then use the appropriate command on your calculator to find the coordinates of the intersection point. From Figure 2 we see that the x-coordinate of this point is x � 8.93.

Plot1 Plot2 Plot3\Y1=50000\Y2=32000 e ˆ (0.05X)\Y3=\Y4=\Y5=\Y6=\Y7=

Figure 1

70,000

00 15

IntersectionX=8.9257421 Y=50000

Figure 2


a. What is an exponential equation?

b. How do logarithms help you solve exponential equations?

c. What is the change-of-base property?

d. Write an application modeled by the equation A � 10,000 �1 � �0.

2

08��

2�5.


Solve each exponential equation. Use a calculator to write the answer in decimal form.

1. 3x � 5 2. 4x � 3 3. 5x � 3 4. 3x � 45. 5�x � 12 6. 7�x � 8 7. 12�x � 5 8. 8�x � 79. 8x�1 � 4 10. 9x�1 � 3 11. 4x�1 � 4 12. 3x�1 � 9

13. 32x�1 � 2 14. 22x�1 � 3 15. 31�2x � 2 16. 21�2x � 317. 153x�4 � 10 18. 103x�4 � 15 19. 65�2x � 4 20. 97�3x � 5

Use the change-of-base property and a calculator to find a decimal approximation toeach of the following logarithms.

21. log8 16 22. log9 27 23. log16 8 24. log27 925. log7 15 26. log3 12 27. log15 7 28. log12 329. log8 240 30. log6 180 31. log4 321 32. log5 462

33. Compound Interest How long will it take for $500 to double if it is invested at6% annual interest compounded twice a year?

34. Compound Interest How long will it take for $500 to double if it is invested at6% annual interest compounded 12 times a year?

35. Compound Interest How long will it take for $1,000 to triple if it is invested at12% annual interest compounded 6 times a year?

36. Compound Interest How long will it take for $1,000 to become $4,000 if it isinvested at 12% annual interest compounded 6 times a year?

37. Doubling Time How long does it take for an amount of money P to doubleitself if it is invested at 8% interest compounded 4 times a year?

38. Tripling Time How long does it take for an amount of money P to triple itselfif it is invested at 8% interest compounded 4 times a year?

39. Tripling Time If a $25 investment is worth $75 today, how long ago must that$25 have been invested at 6% interest compounded twice a year?

40. Doubling Time If a $25 investment is worth $50 today, how long ago must that$25 have been invested at 6% interest compounded twice a year?

Recall that if P dollars are invested in an account with annual interest rate r, com-pounded continuously, then the amount of money in the account after t years is givenby the formula A(t) � Pert.

41. Continuously Compounded Interest Repeat Problem 33 if the interest is com-pounded continuously.

42. Continuously Compounded Interest Repeat Problem 36 if the interest is com-pounded continuously.

43. Continuously Compounded Interest How long will it take $500 to triple if itis invested at 6% annual interest, compounded continuously?

44. Continuously Compounded Interest How long will it take $500 to triple if itis invested at 12% annual interest, compounded continuously?

45. Exponential Growth Suppose that the population in a small city is 32,000 atthe beginning of 1994 and that the city council assumes that the population sizet years later can be estimated by the equation

P(t) � 32,000e0.05t

Approximately when will the city have a population of 64,000?


PROBLEM SET B.5

46. Exponential Growth Suppose the population of a city is given by the equation

P(t) � 100,000e0.05t

where t is the number of years from the present time. How large is the popula-tion now? (Now corresponds to a certain value of t. Once you realize what thatvalue of t is, the problem becomes very simple.)


P(t) � 15,000e0.04t

where t is the number of years from the present time. How long will it take forthe population to reach 45,000? Solve the problem using algebraic methods first,and then verify your solution with your graphing calculator using the intersec-tion of graphs method.


P(t) � 15,000e0.08t

where t is the number of years from the present time. How long will it take forthe population to reach 45,000? Solve the problem using algebraic methods first,and then verify your solution with your graphing calculator using the intersec-tion of graphs method.


49. Solve the formula A � Pert for t.50. Solve the formula A � Pe�rt for t.51. Solve the formula A � P � 2�kt for t.52. Solve the formula A � P � 2kt for t.53. Solve the formula A � P(1 � r)t for t.54. Solve the formula A � P(1 � r)t for t.


exponential and logarithmic functions - … · 2012-12-31 · 466 appendix b exponential and...

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