Logarithmic Functions
as Inverses
7-3
Vocabulary
Review
1. Circle the base in each power.
24
522
3x 5
(x 2 2)3
3x
Vocabulary Builder
logarithm (noun)
uh rith um
LAWG
Related Words: base, power, exponent
Definition: The logarithm base b of a positive number x is the exponent to which
base b must be raised to get x.
Examples: Since 24 5 16, the logarithm base 2 of 16 is 4.
Math Usage: A logarithm is written as log b x and is read “log base b of x.”
Use Your Vocabulary
Write each logarithm in words. The first one is done for you.
Sample: log 2 16
2. log 3 4
log base 2 of 16
3. log 4 3
log base 3 of 4
4. log x y
log base 4 of 3
log base x of y
Key Concept Logarithm
A logarithm base b of a positive number x satisfies the following definition.
log b x 5 y if and only if b y 5 x
Chapter 7
194
5. log y x
log base y of x
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Similarly, since 43 5 64, the logarithm base 4 of 64 is 3.
6. Draw a line from each logarithm equation in Column A to its exponential equation
in Column B.
Column A
Column B
log 2 16 5 4
103 5 1000
log 3 9 5 2
by 5 x
log 10 1000 2 3
32 5 9
log b x 5 y
24 5 16
Problem 1 Writing Exponential Equations in Logarithmic Form
Got It? What is the logarithmic form of the equation 36 5 6 2 ?
7. Use the definition of logarithm: If x 5 b y , then log b x 5 y .
2
Since 36 5 6
, log 6 36 5
2 .
8. The logarithmic form of the equation is
log6 36 5 2
.
Problem 2 Evaluating a Logarithm
Got It? What is the value of log 5 125?
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
9. Use the justifications at the right to complete the equations.
x 5 log 5 125
125 5 5 x
5
Write the logarithm.
Use the definition of a logarithm to write an exponential equation.
3
5 5x
Write 125 as a power of 5.
3
5x
Since the bases are the same, the exponents must be equal.
log 5 125 5
3
Write the value of log 5 125.
A common logarithm is a logarithm with base 10. You can write a common logarithm
log10 x simply as log x, without showing the 10.
Problem 3 Using a Logarithm Scale
Got It? In 1995, an earthquake in Mexico registered 8.0 on the Richter scale. In
2001, an earthquake of magnitude 6.8 shook Washington state. How many times
more intense was the 1995 earthquake than the 2001 earthquake?
I
Use the formula log I1 5 M1 2 M2 to compare the intensity levels of earthquakes,
2
where I is the intensity level and M is the magnitude on the Richter scale.
10. Circle the equation that models the problem.
I1
log I 5 2001 2 1995
2
I1
log I 5 8.0 2 6.8
2
195
I1
log I 5 101.2 2 106
2
Lesson 7-3
11. Simplify the equation you circled in Exercise 10.
I
log I1 5 1.2
2
12. Circle the common logarithm that corresponds to the simplified equation.
I1
1.2
I2 5 10
I1
6
I2 5 10
I1
24.8
I2 5 10
13. Use a calculator. The 1995 earthquake was about 15.85 times more intense than
the 2001 earthquake.
Problem 4
Graphing a Logarithmic Function
Got It? What is the graph of y 5 log 4 x? Describe the domain, range, y-intercept,
and asymptotes.
14. The graph of y 5 4 x is at the right.
y
Circle the graph of its inverse y 5 log 4 x.
y
12
y
24
O
16
8
8
16
8
8
16
x
24
4
x
O
8
16
x
24
12
24
15. Draw a line from each item in Column A to a corresponding item in Column B.
Column A
Column B
Domain of y 5 log 4 x
x50
Range of y 5 log 4 x
x.0
y-intercept of graph of y 5 log 4 x
y.0
Vertical asymptote(s) of y 5 log 4 x
none
all real numbers
Concept Summary Families of Logarithmic Functions
Parent function
y 5 log b x, b . 0, b 2 1
Stretch ( u a u . 1)
Compression (shrink) (0 , u a u , 1) ¶
Reflection (a , 0) in x-axis
y 5 a log b x
Translations (horzontal by h; vertical by k)
y 5 log b (x 2 h) 1 k
All transformations together
y 5 a log b (x 2 h) 1 k
Chapter 7
8
196
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
O
4
16. Circle each equation that is a compression of the parent function. Underline each
equation that is a reflection of the parent function.
y 5 23 log 4 x
y 5 0.2 log8 x
y 5 log 3(x 2 5)
y 5 15 log10 x
Problem 5 Translating y 5 logb x
Got It? How does the graph of y 5 logb (x 2 3) 1 4 compare to
the graph of the parent function?
17. The graph at the right shows the parent function, y 5 log 2 x.
Underline the correct word or value to complete each sentence.
8
y
4
y
฀log2 x
To graph y 5 log 2 (x 2 3) 1 4, translate the graph of the parent
function 2 / 3 / 4 units left / right and 2 / 3 / 4 units up / down .
The domain becomes x . 24 / 3 . The range does / does not change.
x
O
4
8
12
16
4
Lesson Check • Do you UNDERSTAND?
Vocabulary Determine whether each logarithm is a common logarithm.
log2 4
log 64
log10 100
log 5 5
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
18. Circle the base of each logarithm above.
Complete.
19. Common logarithms have a base of 10 .
20. Logarithms without a base are assumed to have a base of 10 .
21. Write Y for yes or N for no to indicate whether each logarithm is a common logarithm.
N log 4
2
Y log 64
Y log 100
10
N log 5
5
Math Success
Check off the vocabulary words that you understand.
logarithm
logarithmic function
common logarithm
logarithmic scale
Rate how well you can use and graph logarithms.
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review
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get it!
10
197
Lesson 7-3