7 logarithmic functionsdustintench.pbworks.com/f/2010+ga+math+3+student... · lesson 7.1 l...

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Chapter 7 l Logarithmic Functions 279

7.1 What’s the Inverse of an Exponent?Logarithmic Functions as

Inverses | p. 281

7.2 Do I Have the Right Form?Exponential and Logarithmic

Forms | p. 289

7.3 It’s All in the GraphGraphs of Logarithmic

Functions | p. 293

7.4 Transformers Again!Transformations of Logarithmic

Functions | p. 301

Logarithmic Functions7CHAPTER

The human ear is capable of hearing sounds across a wide dynamic range. The softest noise the

average human can hear is 0 decibels (dB), which is equivalent to a mosquito flying three meters

away. By comparison, a pop music concert, at about 115 dB, is over 10 billion times louder. The

decibel scale is an example of a logarithmic scale, which can be used to plot very large and very

small numbers along a single axis. You will learn about logarithms and their uses.

280 Chapter 7 l Logarithmic Functions

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Lesson 7.1 l Logarithmic Functions as Inverses 281

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7.1 What’s the Inverse of an Exponent?Logarithmic Functions as Inverses

ObjectivesIn this lesson you will:

l Graph the inverse of exponential

functions.

l Define the inverse of exponential

functions.

l Determine the domain, range, and

asymptotes of the inverse of

exponential functions.

Key Termsl logarithm

l logarithmic function

l common logarithm

l natural logarithm

Problem 1 Graphing the Inverse of Exponential Functions

1. Graph and label the function f(x) � 2x and the line y � x.

86

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2

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6

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2. Complete the tables for the function f(x) � 2x and its inverse. Then plot each

point on the grid from Question 1.

x f( x) � 2x

�3 1 __ 8

�2

�1

0

1

2

3

x f �1( x)

1 __ 8

3. Connect the points of f �1(x) with a smooth curve. Then, label the graph

as f �1(x).

4. Is the inverse f �1(x) a function? Explain.

5. What are the domain and range of the exponential function?

6. What are the domain and range of the inverse of the exponential function?

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7. What do you notice about the domain and range of the exponential function

and its inverse?

8. What is the asymptote of the exponential function?

9. What is the asymptote of the inverse of the exponential function?

10. What do you notice about the asymptotes of the exponential function

and its inverse?

11. Graph and label the function g(x) � 3x and the line y � x.

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12. Complete the tables for the function g(x) � 3x and its inverse. Then plot each

point on the grid from Question 11.

x g( x) � 3x

�3 1 ___ 27

�2

�1

0

1

2

x g�1( x)

13. Connect the points of g�1(x) with a smooth curve. Then, label the graph

as g�1(x).

14. Is the inverse g�1(x) a function? Explain.



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and its inverse?



20. What do you notice about the asymptotes of the exponential function and

its inverse?

21. The graph of the function h( x) � bx and the line y � x are shown. Sketch the

graph of h�1( x).

y

x

y = xh(x)

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22. Is the inverse h�1(x) a function? Explain.




and its inverse?



28. What do you notice about the asymptotes of the exponential function and

its inverse?

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Problem 2 Defining the Inverse of an Exponential Function

In Problem 1, you graphed the inverse of the exponential function h(x) � bx as

a reflection about the line y � x. You also examined the domain, range, and

asymptote of the exponential function and its inverse. You have not encountered

a function with the properties of the inverse of an exponential function. So, it is

necessary to define a new function for the inverse of an exponential function.

The logarithm of a number to a given base is the power or exponent to which

the base must be raised in order to produce the number. For example, if a � bc,

then the logarithm of a to the base b is c. This logarithm is written as logb a � c.

A logarithmic function is a function involving a logarithm.

Logarithms were first conceived by a Swiss clockmaker and amateur mathematician

Joost Bürgi but became more widely known and used after the publication of a

book by Scottish mathematician John Napier in 1614. Logarithms were originally

used to make complex computations in astronomy, surveying, and other sciences

easier and more accurate. With the invention of calculators and computers, the use

of logarithms as a tool for calculation has decreased. However, many real-world

situations can be modeled using logarithmic functions.

The two frequently used logarithms are logarithms with base 10 and base e.

A common logarithm is a logarithm with base 10 and is usually written log without

a base specified. A natural logarithm is a logarithm with base e, Euler’s constant,

and is usually written as In. Many graphing calculators only have keys for common

logarithms and natural logarithms.

1. Graph and label the functions f (x) � log x and g(x) � In x using a

graphing calculator.

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2. What is the inverse of the logarithmic function f(x) � log x?

3. What is the inverse of the logarithmic function f(x) � In x?

Be prepared to share your methods and solutions.

Lesson 7.2 l Exponential and Logarithmic Forms 289

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7.2 Do I Have the Right Form?Exponential and Logarithmic Forms


l Convert from exponential equations to logarithmic equations.

l Convert from logarithmic equations to exponential equations.

l Evaluate logarithmic expressions.

Problem 1 Converting Between Exponential and Logarithmic Forms

Remember that the definition of a logarithm allows you to convert the exponential

equation a � bc to the logarithmic equation logb a � c.

1. Write each exponential equation as a logarithmic equation using the definition

of logarithms.

a. 23 � 8 b. 5�2 � 1 ___ 25

c. 104 � 10,000 d. 122 � 144

e. ( 1 __ 3 )

�4

� 81 f. ( 1 __ 5 )

4

� 1 ____ 625

2. Write each logarithmic equation as an exponential equation using the

definition of logarithms.

a. log2 16 � 4 b. log

3 1 ___ 27

� �3

c. log 0.0001 � �4 d. loga b � c

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Problem 2 Evaluating Logarithmic Expressions

1. Evaluate each logarithmic expression without using a calculator. Explain how

you calculated each.

a. log2 1024 b. log

5 1 ___ 25

c. log7 343 d. log 100,000

e. log4 2 f. log

3 1 ___ 27

g. log5

3 �� 25 h. log 7 �� 100,000

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Lesson 7.2 l Exponential and Logarithmic Forms 291

7

2. Evaluate each logarithmic expression. Use a calculator if necessary.

a. log 100 b. log 0.01

c. log 7 d. log 70

e. log 700 f. log 0.07

g. log 343


a. In 100 b. In 10

c. In 25 d. In 0.25

e. In 0.004 f. In e

g. In e 2 __ 3 h. In 6.25

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a. log 64 b. log 8

c. 2 log 8 d. In 12

e. In 4 f. In 3

g. log 36 h. log 6

i. In 5 __ 3 j. In 5

k. In 3


Lesson 7.3 l Graphs of Logarithmic Functions 293

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7.3 It’s All in the GraphGraphs of Logarithmic Functions


l Graph logarithmic functions.

l Determine the characteristics of logarithmic functions.

Problem 1 The Extraordinary Graph of an Exponential Function

1. Graph the function f (x) � 10x for x-values between �5 and 5 and y-values

between 0 and 100.

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between 0 and 1000.


between 0 and 10,000.

4. Describe the similarities and differences between the graphs in

Questions 1 through 3.

5. Describe how the scale of the x-axis changes from each graph.

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6. Describe how the scale of the y-axis changes from each graph.

7. Imagine a very large sheet of graph paper with every square grid measuring

one tenth of an inch. You set the scale on both the x-axis and the y-axis at

one unit. Describe the coordinates of each point on the graph of the function

f (x) � 10x and the point’s distance from the origin on the graph paper.

a. A point one inch above the x-axis.

b. A point one foot above the x-axis.

c. A point one hundred feet above the x-axis.

d. A point one mile above the x-axis.

e. A point one foot to the right of the origin.

As you can see in Problem 1, the y-values of an exponential function increase very

rapidly, even for relatively small changes in the values of x.

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Problem 2 The Extraordinary Graph of a Logarithmic Function

1. Graph the function f (x) � log x for x-values between 0 and 100 and y-values

between �5 and 5.

2. Graph the function f (x) � log x for x-values between 0 and 1000 and y-values

between �10 and 10.

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3. Graph the function f (x) � log x for x-values between 0 and 10,000 and y-values

between �20 and 20.

4. Describe the similarities and differences between the graphs in

Questions 1 through 3.

5. Describe how the scale of the x-axis changes from each graph.

6. Describe how the scale of the y-axis changes from each graph.

7. Imagine a very large sheet of graph paper with every square grid measuring

one tenth of an inch. You set the scale on both the x-axis and the y-axis at

one unit. Describe the coordinates of each point on the graph of the function

f (x) � log x and the point’s distance from the origin on the graph paper.

a. A point one inch along the x-axis.

b. A point one foot along the x-axis.

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c. A point one hundred feet along the x-axis.

d. A point one mile along the x-axis.

e. A point one foot above the x-axis.

As you can see in Problem 2, the y-values of a logarithmic function increase very

slowly, even for relatively large values of x.

Problem 3 Logarithmic Functions 1. The graph of f (x) � log

a x is shown. Label the coordinates of each of the

three points.

x

–1

–2

–3

y

–4

4

3

2

1

f(x) = loga x

2. What are the domain and range of f (x) � loga x ?

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3. What are the intercepts of f (x) � loga x?

4. What are the asymptotes of f (x) � loga x?

5. For what x-values is the function f (x) � loga x increasing or decreasing?


Lesson 7.4 l Transformations of Logarithmic Functions 301

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7.4 Transformers Again!Transformations of Logarithmic Functions

ObjectiveIn this lesson you will:

l Transform logarithmic functions algebraically and graphically.

Problem 1 Horizontal and Vertical Translations

Earlier you learned that the graph of a function f (x) is translated vertically k units if a

constant k is added to the equation: f (x) � k.

1. Sketch and label the graphs of f (x) � log x, f (x) � log x � 3,

and f (x) � log x � 4.

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The graph of a function f (x) is translated horizontally h units if a constant h is

subtracted from the variable in the function: f (x � h).

2. Sketch and label the graphs of f (x) � log x, f (x) � log(x � 3), and

f (x) � log (x � 4).

3. Sketch and label the graph of f (x) � In(x � 3) � 5 using the graph of

f (x) � In x.

x86

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Problem 2 ReflectionsEarlier you learned that the graph of a function f (x) is reflected about the x-axis if

the equation of the function is multiplied by �1: �f (x).

1. Sketch and label the graph of f (x) � �In x using the graph of f (x) � In x.

x86

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4

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8

–2–2

4 1412102

–4

–6

–8

y

f(x) = In x

Earlier you learned that the graph of a function f (x) is reflected about the y-axis if

the argument of the equation of the function is multiplied by �1: f (�x) .

2. Sketch and label the graph of f (x) � log(�x) using the graph of f (x) � log x.

x42

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–4

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f(x) = log x

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3. Sketch and label the graph of f (x) � �In(�x) using the graph of f (x) � In x.

x42

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–2 86–4

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–6

–8

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y

f(x) = In(x)

Problem 3 DilationsEarlier you learned that the graph of a function f (x) is dilated a units vertically if the

function is multiplied by a constant a: af (x).

1. Sketch and label the graphs of f (x) � log x, f (x) � 2 log x, and f (x) � 1 __ 2 log x.

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The graph of a function f (x) is dilated horizontally c units if the argument of the

function is multiplied by the constant c: f (cx).

2. Sketch and label the graphs of f (x) � log x, f (x) � log 3x, and f (x) � log 1 __ 3 x.

Problem 4 Putting It All Together! 1. Sketch and label the graph of f (x) � 2 In x � 3 using the graph of f (x) � In x.

x42

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–2

–3

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f(x) = In x

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2. Sketch and label the graph of f (x) � �In(x � 2) � 3 using the graph

of f (x) � In x.

x42

1

2

3

4

–2–1

86 10 1412

–2

–3

–4

y

f(x) = In x

3. Sketch and label the graph of f (x) � 2 log(�(x � 2)) using the graph

of f (x) � log x.

x42

1

2

3

4

–8–1

–2 86–4

–2

–3

–4

–6

y

f(x) = log x


7 logarithmic functionsdustintench.pbworks.com/f/2010+ga+math+3+student... · lesson 7.1 l...

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