Holt Algebra 2 7-3 Logarithmic Functions Write equivalent forms for exponential and logarithmic...

Holt Algebra 2

7-3 Logarithmic Functions

Write equivalent forms for exponential and logarithmic functions.

Write, evaluate, and graph logarithmic functions.

Objectives & Vocabulary

logarithm common logarithmlogarithmic function

Holt Algebra 2


Exponential Equation

Logarithmic Form

92= 81

33 = 27

x0 = 1(x ≠ 0)

Notes #1-2 Rewrite in other form

a.

b.

c.

Logarithmic Form


log101000 = 3

log12144 = 2

log 8 = –3

3A. Change 64 = 1296 to logarithmic form

B. Change log279 = to exponential form.23

Calculate (without a calculator).

4. log864 5. log3 127

Holt Algebra 2


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

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

Reading Math

Holt Algebra 2


Example 1: Converting from Logarithmic to Exponential Form

Write each logarithmic form in exponential equation.

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

The logarithm is the exponent.

A logarithm can be a negative number.Any nonzero base to the zero power is 1.

Logarithmic Form


log99 = 1

log2512 = 9

log82 =

log4 = –2

logb1 = 0

116

13

91 = 9

29 = 512

138 = 2

1164–2 =

b0 = 1

Holt Algebra 2


Write each exponential equation in logarithmic form.

Example 2: Converting from Exponential to Logarithmic Form

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

The exponent is the logarithm.

An exponent (or log) can be negative.

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


Logarithmic Form

35 = 243

25 = 5

104 = 10,000

6–1 =

ab = c

16

12

log3243 = 5

12log255 =

log1010,000 = 4

16log6 = –1

logac =b

Holt Algebra 2


Write each exponential equation in logarithmic form.

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

The exponent of the logarithm.

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


Logarithmic Form

92= 81

33 = 27

x0 = 1(x ≠ 0)

Notes #1

a.

b.

c.

log981 = 2

log327 = 3

logx1 = 0

Holt Algebra 2


Write each logarithmic form in exponential equation.

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

The logarithm is the exponent.

An logarithm can be negative.

Logarithmic Form


log101000 = 3

log12144 = 2

log 8 = –312

Notes #2

103 = 1000

122 = 144

12

–3

= 8

Holt Algebra 2


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.

Holt Algebra 2


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 = log105.

You can use mental math to evaluate some logarithms.

Holt Algebra 2


Evaluate by without a calculator.

Example 3A: Evaluating Logarithms by Using Mental Math

The log is the exponent.

Think: What power of 10 is 0.01?

log 0.01

10? = 0.01

10–2 = 0.01

log 0.01 = –2

Holt Algebra 2


Evaluate without a calculator.

Example 3B: Evaluating Logarithms by Using Mental Math

The log is the exponent.

log5 125

5? = 125

log5125 = 3

Holt Algebra 2


Evaluate without a calculator.

Example 3C/3D: Evaluating Logarithms by Using Mental Math

3c. log5

15

log5

= –115

3d. log25

0.04

log25

0.04 = –1

Holt Algebra 2


Because logarithms are the inverses of exponents, the inverse of an exponential function, such as y = 2x, is a logarithmic function, such as y = log2x.

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

The domain of y = 2x is all real numbers (R), and the range is {y|y > 0}. The domain of y = log2x is {x|x > 0}, and the range is all real numbers (R).

Holt Algebra 2


Example 4A: Graphing Logarithmic Functions

x –2 –1 0 1 2

f(x) =( ) x 4 2 1

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

1 2

1 2

1 2

1 4

f(x) = x 1 2

Use the x-values {–2, –1, 0, 1, 2}. Graph the function and its inverse. Describe the domain and range of the inverse function.

Holt Algebra 2


The domain of f–1(x) is {x|x > 0}, and the range is R.

To graph the inverse, f–1(x) = log x, by using a table of values.

1 2

1 2

1 4

1 2

x 4 2 1

f –1(x) =log x –2 –1 0 1 2

Example 4A Continued

Holt Algebra 2


Notes (continued)

3A. Change 64 = 1296 to logarithmic form log61296 = 4

B. Change log279 = to exponential form.23

27 = 923

4. log864

5. log3

Calculate the following using mental math (without a calculator).

127

2

–3

Holt Algebra 2


6. Use the x-values {–1, 0, 1, 2} to graph

f(x) = 3x Then graph its inverse. Describe the

domain and range of the inverse function.

Notes (graphing)

D: {x > 0}; R: all real numbers

Holt Algebra 2 7-3 Logarithmic Functions Write equivalent forms for exponential and logarithmic...

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