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How many times would you have to double $1 before you had $8? You could use an exponential equation to model this situation. 1 ( 2 x ) = 8. You may be able to solve this equation by using mental math if you know that 2 3 = 8. So you would have to double the dollar 3 times to have $8.

How many times would you have to double $1 to have $512? You could solve this problem if you could solve 2 x = 8 by using an inverse operation that undoes raising a base to an exponent. This operation is called finding the logarithm. A logarithm is the exponent to which a specified base is raised to obtain a given value.

You can write an exponential equation as a logarithmic equation and vice versa.

Exponential Equation

b > 0, b ≠ 1

Logarithmic Equation

1E X A M P L E Converting from Exponential to Logarithmic Form

Write each exponential equation in logarithmic form.

Exponential Equation

Logarithmic Form

a. 2 6 = 64 log 2 64 = 6

b. 4 1 = 4 log 4 4 = 1

c. 5 0 = 1 log 5 1 = 0

d. 5 -2 = 0.04 log 5 0.04 = -2

e. 3 x = 81 log 3 81 = x

The base of the exponent becomes the base of the logarithm.

The exponent is the logarithm.

Any nonzero base to the 0 power is 1.

An exponent (or log) can be negative.

The log (and the exponent) can be a variable.

Write each exponential equation in logarithmic form.

1a. 9 2 = 81 1b. 3 3 = 27 1c. x 0 = 1 (x ≠ 0)

7-3 Logarithmic Functions

ObjectivesWrite equivalent forms for exponential and logarithmic functions.

Write, evaluate, and graph logarithmic functions.

Vocabularylogarithmcommon logarithmlogarithmic function

Why learn this?A logarithmic scale is used to measure the acidity, or pH, of water. (See Example 5.)

Read lo g b a = x, as “the log base b of a is x.” Notice that the log is the exponent.

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TEKS 2A.11.A Exponential and logarithmic functions: develop the definition of logarithms by … describing the relationship between exponential functions and their inverses.

Also 2A.4.A, 2A.11.C

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506 Chapter 7 Exponential and Logarithmic Functions

2E X A M P L E Converting from Logarithmic to Exponential Form

Write each logarithmic equation in exponential form.

Logarithmic Equation

Exponential Form

a. log 10 100 = 2 10 2 = 100

b. log 7 49 = 2 7 2 = 49

c. log 8 0.125 = -1 8 -1 = 0.125

d. log 5 5 = 1 5 1 = 5

e. log 12 1 = 0 12 0 = 1

The base of the logarithm becomes the base of the power.

The logarithm is the exponent.

A logarithm can be a negative number.

Write each logarithmic equation in exponential form.

2a. log 10 10 = 1 2b. log 12 144 = 2 2c. log 1 _ 2

8 = -3

A logarithm is an exponent, so the rules for exponents also apply to logarithms. You may have noticed the following properties in the last example.

For any base b such that b > 0 and b ≠ 1,

LOGARITHMIC FORM EXPONENTIAL FORM EXAMPLE

Logarithm of Base b

lo g b b = 1 b 1 = b

lo g 10 10 = 1 10 1 = 10

Logarithm of 1

lo g b 1 = 0 b 0 = 1lo g 10 1 = 0 10 0 = 1

Special Properties of Logarithms

A logarithm with base 10 is called a common logarithm . If no base is written for a logarithm, the base is assumed to be 10. For example, log 5 = log 10 5.

You can use mental math to evaluate some logarithms.

3E X A M P L E Evaluating Logarithms by Using Mental Math

Evaluate by using mental math.

A log 1000 B log 4 1 _ 4

10 ? = 1000 The log is the exponent. 4 ? = 1 _ 4

10 3 = 1000 Think: What power of 4 -1 = 1 _ 4

the base is the value?

log 1000 = 3 log 4 1 _ 4

= -1

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7- 3 Logarithmic Functions 507

Evaluate by using mental math.

3a. log 0.00001 3b. log 25 0.04

Because logarithms are the inverses of exponents, the inverse of an exponential function, such as y = 2 x , is a logarithmic function , such as y = log 2 x.

You may notice that the domain and range of each function are switched.

The domain of y = 2 x is all real numbers (�), and the range is

⎧ ⎨

⎩ y | y > 0

⎫ ⎬

⎭ . The domain of y = log 2 x is

⎧ ⎨

⎩ x | x > 0

⎫ ⎬

⎭ , and the range is all real numbers (�).

4E X A M P L E Graphing Logarithmic Functions

Use the given x-values to graph each function. Then graph its inverse. Describe the domain and range of the inverse function.

A f (x) = 3 x ; x = -2, -1, 0, 1, and 2

Graph f (x) = 3 x by using a table of values.

x -2 -1 0 1 2

f (x) = 3 x 1 _ 9 1 _

3 1 3 9

To graph the inverse, f -1 (x) = log 3 x, reverse each ordered pair.

x 1 _ 9 1 _

3 1 3 9

f -1 (x) = log 3 x -2 -1 0 1 2

The domain of f -1 (x) is ⎧ ⎨

⎩ x | x > 0

⎫ ⎬

⎭ , and the range is �.

B f (x) = 0. 8 x ; x = -3, 0, 1, 4, and 7

Graph f (x) = 0. 8 x by using a table of values. Round the output values to the nearest tenth, if necessary.

x -3 0 1 4 7

f (x) = 0.8 x 2 1 0.8 0.4 0.2

To graph f -1 (x) = log 0.8 x, reverse each ordered pair.

x 2 1 0.8 0.4 0.2

f -1 (x) = log 0.8 x -3 0 1 4 7

The domain of f -1 (x) is ⎧ ⎨

⎩ x | x > 0

⎫ ⎬

⎭ , and the range is �.

4. Use x = -2, -1, 1, 2, and 3 to graph f (x) = ( 3 __ 4 ) x . Then

graph its inverse. Describe the domain and range of the inverse function.

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508 Chapter 7 Exponential and Logarithmic Functions

5E X A M P L E Environmental Application

Chemists regularly test rain samples to determine the rain’s acidity, or concentration of hydrogen ions ( H + ) . Acidity is measured in pH, as given by the function pH = - log ⎡ ⎣ H + ⎤ ⎦ , where ⎡ ⎣ H + ⎤ ⎦ represents the hydrogen ion concentration in moles per liter.

Find the pH of rainwater from each location.

A Central New Jersey

The hydrogen ion concentration is 0.0000316 moles per liter.

pH = - log ⎡ ⎣ H + ⎤ ⎦

pH = - log (0.0000316) Substitute the known values in the function.

Use a calculator to find the value of the logarithm in base 10. Press the key.

The rainwater has a pH of about 4.5.

B Central North Dakota

The hydrogen ion concentration is 0.0000009 moles per liter.

pH = - log ⎡ ⎣ H + ⎤ ⎦

pH = - log (0.0000009) Substitute the known values in the function.

Use a calculator to find the value of the logarithm in base 10. Press the key.

The rainwater has a pH of about 6.0.

5. What is the pH of iced tea with a hydrogen ion concentration of 0.000158 moles per liter?

THINK AND DISCUSS1. Explain why log b b is always equal to 1 for b > 0 and b ≠ 1.

2. Explain whether log b a is the same as log a b. Support your answer.

3. GET ORGANIZED Copy and complete the graphic organizer. Use your own words to explain a logarithmic function.

Central California0.0000032 mol/L

Central New Jersey0.0000316 mol/L

Eastern Texas0.0000192 mol/L

Eastern Ohio0.0000629 mol/L

Central North Dakota0.0000009 mol/L

Hydrogen Ion Concentrationof Rainwater

The key is used to evaluate logarithms in base

10. is used to find 10 x , the inverse of log.

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