LESSON
15.1
Name
Defining and
Evaluating a
Logarithmic Function
Class
15.1 Defining and Evaluating
a Logarithmic Function
Essential Question: What is the inverse of the exponential function f(x) = b x where b > 0 and
b ≠ 1, and what is the value of f -1(b m) for any real number m?
Common Core Math Standards
Explore
The student is expected to:
F.BF.5(+)

Mathematical Practices
MP.2 Reasoning
Understanding Logarithmic Functions as Inverses
of Exponential Functions
Graph f -1(x) = log 2 x using the graph of ƒ(x) = 2 x shown. Begin by reflecting the labeled
points on the graph of ƒ(x) = 2 x across the line y = x and labeling the reflected points with
their coordinates. Then draw a smooth curve through the reflected points.
y
Language Objective
6
(2, 4)
© Houghton Mifflin Harcourt Publishing Company
(1, 2) 4
(0, 1)
2
(- 1,0.5)
(- 2,0.25)
Possible answer: The inverse of f (x) = b x is
f -1(x) = log b x, the logarithm with base b of x. The
value of f -1(b m) is m because the inverse function
accepts a power of b as an input and delivers the
exponent as an output.
y=x
(3, 8)
8
Discuss with a partner the relationship between exponential and
logarithmic functions.
Essential Question: What is the
inverse of the exponential function
f (x) = b x where b > 0 and b ≠ 1, and
what is the value of f -1(b m) for any real
number m?
Resource
Locker
An exponential function such as ƒ(x) = 2 x accepts values of the exponent as inputs and delivers the corresponding
power of 2 as the outputs. The inverse of an exponential function is called a logarithmic function. For ƒ (x) = 2 x, the
inverse function is written f -1(x) = log 2 x, which is read either as “the logarithm with base 2 of x” or simply as “log
base 2 of x.” It accepts powers of 2 as inputs and delivers the corresponding exponents as outputs.
Understand the inverse relationship between exponents and logarithms
and use this relationship to solve problems involving logarithms and
exponents. Also F.IF.7, F.IF.7e, F.IF.2
ENGAGE
Date
-4
(2, 1)
(1, 0)
0
-2
(8, 3)
(4, 2)
2
4
(0.5, -1)
(0.25, -2)
-2
x
6
8
-4

Using the labeled points on the graph of f -1(x), complete the following statements.
f -1(0.25) = log 2 0.25 = -2
f -1(0.5) = log 2 0.5 = -1
f -1(1) = log 2 1 = 0
f -1(2) = log 2 2 = 1
f -1(4) = log 2 4 = 2
f -1(8) = log 2 8 = 3
Module 15
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EDIT--Chan
DO NOT Key=NL-B;CA-B
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Lesson 1
745
gh "File info"
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Date
Class
Resource
b > 0 and
Locker
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ential functi
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F.IF.2
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F.IF.7, F.IF.7e,
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Inverses
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F.BF.5(+)
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ing
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ponding
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Underst ntial Functions
s the corres
= 2 , the
and deliver
Explore
For ƒ (x)
ent as inputs
function.
of Expone
as “log
of the expon
logarithmic 2 of x” or simply
x
accepts values
is called a
base
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ƒ(x) = 2
hm with
s.
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exponential
“the logarit
as output
either as
ential functio s. The inverse of an
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An expon
which is read the corresponding
s
2 as the output f -1 (x) = log 2 x,
power of
and deliver
labeled
is written
inputs
n
the
as
ing
2
reflect
inverse functio accepts powers of
. Begin by
points with
x.” It
2x shown
reflected
base 2 of
of ƒ(x) =
labeling the
Quest
Essential

6
(8, 3)
(2, 4)
(1, 2) 4
(4, 2)
(2, 1)
(0, 1)
2
(1, 0)
(- 1,0.5)
)
8
(- 2,0.25
6
4
2
0
(0.5, -1)
-4 -2
-2 (0.25, -2)
x
of
on the graph
labeled points
= -2
) = log 2 0.25
f -1 (0.25
= -1
log 2 0.5
f -1 (0.5) =
= 0
log 2 1
f -1 (1) =
= 2
log 2 4
© Houghto
f -1 (8) =
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ents.
ing statem
= 1
log 2 2
f -1 (4) =
A2_MNLE
ete the follow
f -1 (x), compl
Using the
n Mifflin

f -1 (2) =
Lesson 15.1
Turn to these pages to
find this lesson in the
hardcover student
edition.
-4
Module 15
745
HARDCOVER PAGES 541552
graph
.
= x and
x using the x
ed points
the line y
-1 (x) = log 2
2 across
h the reflect
Graph f
of ƒ(x) =
h curve throug
the graph
draw a smoot
points on
y=x
y
inates. Then
their coord
(3, 8)
8
y
g Compan
A2_MNLESE385900_U6M15L1.indd 745
Harcour t
View the Engage section online. Discuss the photo
and how air pressure is a function of altitude. Then
preview the Lesson Performance Task.
aluating
ing and Ev
n
15.1 Defingarithmic Functio
a Lo
Name
Publishin
PREVIEW: LESSON
PERFORMANCE TASK
745
= 3
log 2 8
Lesson 1
745
8/20/14
6:05 PM
8/20/14 6:04 PM
Reflect
1.
EXPLORE
Explain why the domain of ƒ (x) = 2 x doesn’t need to be restricted in order for its inverse to be a function.
The exponential function f (x) = 2 x is a one-to-one function, so its inverse is a function.
2.
State the domain and range of f -1(x) = log 2 x using set notation.
The domain is {x|x > 0}, and the range is {y|-∞ < y < +∞}.
3.
Identify any intercepts and asymptotes for the graph of f -1(x) = log 2 x.
The graph has an x-intercept at x = 1 and no y-intercepts. The y-axis (x = 0) is the graph’s
Understanding Logarithmic
Functions as Inverses of Exponential
Functions
INTEGRATE TECHNOLOGY
Students have the option of completing the Explore
activity either in the book or online.
asymptote.
4.
Is f -1(x) = log 2 x an increasing function or a decreasing function?
The function is an increasing function.
5.
How does f (x) = log 2 x behave as x increases without bound? As x decreases toward 0?
As x → + ∞, f -1(x) → +∞. As x → 0 +, f -1(x) → - ∞.
6.
Based on the inverse relationship between ƒ(x) = 2 x and ƒ -1(x) = log 2 x, complete this statement:
QUESTIONING STRATEGIES
-1
f -1 (16) = log 2 16 = 4 because f
Explain 1
Why does reflecting the graph of the
exponential function across the line y = x
produce the graph of a logarithmic
function? Because the graphs of inverse functions
are reflections across the line y = x, and the inverse
of an exponential function is a logarithmic function.
( 4 ) = 16 .
Converting Between Exponential and Logarithmic
Forms of Equations
In general, the exponential function f (x) = b x, where b > 0 and b ≠ 1, has thex logarithmic function f -1 (x) = log b x
as its inverse. For instance, if f (x) = 3 x, then f -1 (x) = log 3 x, and if f (x) = __14 , then f -1 (x) = log __1 x. The inverse
4
relationship between exponential functions and logarithmic functions also means that you can write any exponential
equation as a logarithmic equation and any logarithmic equation as an exponential equation.
Exponential Equation
© Houghton Mifflin Harcourt Publishing Company
()
Logarithmic Equation
b =a
log b a = x
x
b > 0, b ≠ 1
Module 15
746
How do you know that the graph of the
logarithmic function has no horizontal
asymptotes? The graph of the exponential function
has no vertical asymptotes.
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Communication
MP.3 Have students discuss the attributes of the
graph of the logarithmic function. Ask them to
identify the domain and range of the function, and
any intercepts. Also have them determine whether
the function is increasing or decreasing, and describe
its end behavior. Have them compare these attributes
to those of the graph of the exponential function.
Lesson 1
PROFESSIONAL DEVELOPMENT
A2_MNLESE385900_U6M15L1.indd 746
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Learning Progressions
Students have learned about functions and their inverses in previous lessons. They
have also learned about exponential functions. In this lesson, students learn that
the inverse of an exponential function is called a logarithmic function. They learn
how to evaluate logarithmic functions, and how to work with logarithmic
functions that model real-world situations. In the following lesson, students will
learn to how to graph logarithmic functions.
Defining and Evaluating a Logarithmic Function
746
Example 1
EXPLAIN 1
Complete the table by writing each given equation in its alternate form.

Exponential Equation
Converting Between Exponential and
Logarithmic Forms of Equations
Logarithmic Equation
4 = 64
?
?
1 = -2
log 5 _
25
3
(_32 ) = q
p
QUESTIONING STRATEGIES
?
log _1 m = n
?
In the expression log b y = x, what does b
represent? b is the base in the power b x.
2
Think of each equation as involving an exponential function or a logarithmic function. Identify the function’s
base, input, and output. For the inverse function, use the same base but switch the input and output.
In the expression log b y = x, what does x
represent? x is the exponent in the power b x.
Think of the equation 4 3 = 64 as involving an exponential function with base 4. The input is 3, and the
output is 64. So, the inverse function (a logarithmic function) also has base 4, but its input is 64, and its
output is 3.
In the equation log b y = x, what does y
represent? y is the value of b x in the
equation b x = y.
1
1
Think of the equation log 5 __
= -2 as involving a logarithmic function with base 5. The input is __
, and
25
25
the output is -2. So, the inverse function (an exponential function) also has base 5, but its input is -2,
1
and its output is __
.
25
()
p
Think of the equation __23 = q as involving an exponential function with base __23 . The input is p, and the
output is q. So, the inverse function (a logarithmic function) also has base __23 , but its input is q, and its
output is p.
Think of the equation log _1 m = n as involving a logarithmic function with base __12 . The input is m, and the
2
output is n. So, the inverse function (an exponential function) also has base __12 , but its input is n, and its
output is m.
© Houghton Mifflin Harcourt Publishing Company
Exponential Equation
Logarithmic Equation
4 3 = 64
log 4 64 = 3
1
5 -2 = _
25
1 = -2
log 5 _
25
(_23 ) = q
(_21 ) = m
p
n
log _2 q = p
3
log _1 m = n
Module 15
2
747
Lesson 1
COLLABORATIVE LEARNING
A2_MNLESE385900_U6M15L1.indd 747
Peer-to-Peer Activity
Ask students, “What are some powers you know by heart?” Write some of these
on one side of the board. Explain that for every power they know, they already
know the logarithm. Write the logarithmic form alongside each of their
exponential equations, with assistance from students. Review the meanings of
positive and negative exponents. Be sure that students are comfortable moving
back and forth between exponential and logarithmic forms.
747
Lesson 15.1
8/20/14 6:13 PM
B
Exponential Equation
Logarithmic Equation
4 -3 =
1
_
64
(_34 ) = s
(_)
1
5
Help students to make a connection between the base
of an exponent and the base of the related logarithm,
and transfer their understanding of the base of an
exponent to its use with logarithms. This will make it
easier for them to convert between forms.
1 = -3
log 4 _
64
r
w
CONNECT VOCABULARY
log 3 243 = 5
3 5 = 243
log _3 s = r
4
=v
log _1 v = w
5
Think of the equation 3 5 = 243 as involving an exponential function with base 3. The input is 5 ,
and the output is 243 . So, the inverse function (a logarithmic function) also has base 3, but its
input is 243 , and its output is 5 .
1
= -3 as involving a logarithmic function with base 4 . The input
Think of the equation log 4 __
64
1
is __
, and the output is -3 . So, the inverse function (an exponential function) also has base 4 , but
64
1
its input is -3 , and its output is __
.
64
()
r
Think of the equation __34 = s as involving an exponential function with base
3
__
3
__
4
and the output is s. So, the inverse function (a logarithmic function) also has base
and its output is r .
Think of the equation log _1 v = w as involving a logarithmic function with base
5
r ,
. The input is
4
1
__
5
, but its input is s,
v
. The input is
1
__
and the output is w . So, the inverse function (an exponential function) also has base 5 , but its
,
input is w , and its output is v .
Reflect
7.
© Houghton Mifflin Harcourt Publishing Company
A student wrote the logarithmic form of the exponential equation 5 0 = 1 as log 5 0 = 1. What did the
student do wrong? What is the correct logarithmic equation?
The student forgot to switch the input and output when writing the logarithmic equation.
The correct equation is log 5 1 = 0.
Your Turn
8.
Complete the table by writing each
given equation in its alternate form.
Exponential Equation
10 4 = 10,000
2 -4 =
1
_
16
(_25 ) = d
c
(_1 )
y
3
Module 15
748
=x
Logarithmic Equation
log 10 10,000 = 4
1 = -4
log 2 _
16
log _2 d = c
5
log _1 x = y
3
Lesson 1
DIFFERENTIATE INSTRUCTION
A2_MNLESE385900_U6M15L1.indd 748
3/24/14 2:05 PM
Auditory Cues
The following rhyme may help students remember how to rewrite an exponential
equation in logarithmic form.
To convert it to the log form,
Remember each component.
The base goes at the bottom,
And the log is the exponent!
Defining and Evaluating a Logarithmic Function
748
Evaluating Logarithmic Functions by Thinking
in Terms of Exponents
Explain 2
EXPLAIN 2
The logarithmic function ƒ(x) = log b x accepts a power of b as an input and delivers an exponent as an output.
In cases where the input of a logarithmic function is a recognizable power of b, you should be able to determine the
function’s output. You may find it helpful first to write a logarithmic equation by letting the output equal y and then
to rewrite the equation in exponential form. Once the bases on each side of the exponential equation are equal, you
can equate their exponents to find y.
Evaluating Logarithmic Functions by
Thinking in Terms of Exponents
Example 2
QUESTIONING STRATEGIES

When evaluating a logarithm such as log 2 x for
some specified value of x, what are you trying
to find? the exponent to which you raise 2 to get x
_
If ƒ (x) = log 10 x, find ƒ(1000), ƒ(0.01), and ƒ(√10 ) .
ƒ(0.01) = y
ƒ (√10 ) = y
log 10 1000 = y
log 10 0.01 = y
log 10 √10 = y
_
10 y = 0.01
10 y = √ 10
10 y = 10 3
10 y = 10 -2
10 y = 10 2
y=3
y = -2
So, ƒ(0.01) = -2.
( )
1
_
So,
1
y=_
2
1.
=_
2
_
ƒ (√10 )
_
1
If ƒ (x) = log _1 x, find ƒ (4), ƒ __
and ƒ (2√2 ).
32
2
ƒ (4) = y
log _1 4 = y
2
()
(_21 ) = 2
(_21 ) = (_21 )
y
1 =4
_
2
y
© Houghton Mifflin Harcourt Publishing Company
_
10 y = 1000
So, ƒ(1000) = 3.

_
ƒ(1000) = y
y
( )
_
ƒ (2√2 ) = y
1 =y
ƒ _
32
1 =y
log _1 _
2 32
_
log _1 2√2 = y
2
()
1
(_12 ) = _
2
1 = _
_
( 2 ) ( 21 )
y
1 =_
1
_
2
32
2
y
5
-2
y = -2
So, ƒ (4) = -2 .
Module 15
y
5
y= 5
( )
1 =
So, ƒ _
32
5 .
(_12 ) = 2 2
(_21 ) = √2 ⋅ 2
(_12 ) = √2
_
(_12 ) = 2
_
(_12 ) = (_21 )
y
_
√
y
_
y
y
y
2
_
3
3
2
-
3
2
3
y = -_
2
_
3
So ƒ(2√2 ) = -_ .
2
749
Lesson 1
LANGUAGE SUPPORT
A2_MNLESE385900_U6M15L1.indd 749
Communicating Math
Give each pair of students an equation of an exponential function, and ask them
to work together to graph it. Then ask them to decide on what the inverse, or
logarithmic function, is, write the equation, and graph it.
749
Lesson 15.1
8/20/14 6:18 PM
Your Turn
9.
( )
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Communication
MP.3 As students work through these problems,
1 , and ƒ(√_
If ƒ(x) = log 7 x, find ƒ (343), ƒ _
7 ).
49
f (343) = y
f
log7 343 = y
7 y = 343
7y = 73
y=3
1
=y
(_
49 )
log7
1
_
=y
49
1
7 =_
49
1
7 =_
y
y
So, f (343) = 3 .
72
2
7y = 7
y = -2
_
f (√7 ) = y
―
log7 √7 = y
―
7 y = √7
1
_
2
have them verbalize their thought processes. This will
enable you to check for understanding, and will also
provide an opportunity for students who are
struggling to hear how other students are thinking
about these problems.
7y = 7
1
y=_
2
1.
So, f (√7 ) = _
2
―
1
So, f (_) = -2 .
49
Explain 3
AVOID COMMON ERRORS
Evaluating Logarithmic Functions Using
a Scientific Calculator
When evaluating logarithmic expressions, students
sometimes confuse logs that are negative numbers
with those that are fractions. For example, when
1
, students may think the
asked to evaluate log 4 __
16
value must be a fraction, since the argument is a
fraction. Help them to see that the value must instead
be a negative number, in this case -2, in order for 4 2
to be in the denominator of the fraction.
You can use a scientific calculator to find the logarithm of any positive number x when the logarithm’s base is either
10 or e. When the base is 10, you are finding what is called the common logarithm of x, and you use the calculator’s
LOG
key because log 10 x is also written as log x (where the base is understood to be 10). When the base is e, you are
finding what is called the natural logarithm of x, and you use the calculator’s LN key because log e x is also written
as ln x.
Example 3

Use a scientific calculator to find the common logarithm and the natural
logarithm of the given number. Verify each result by evaluating the
appropriate exponential expression.
13
A2_MNLESE385900_U6M15L1.indd 750
Evaluating Logarithmic Functions
Using a Scientific Calculator
QUESTIONING STRATEGIES
Why does the calculator return an error
message when you enter log (-1)? You
cannot take the log of a negative number. There is
no number to which 10 can be raised to get -1.
So, ln 13 ≈ 2.565.
So, log 13 ≈ 1.114.
Module 15
EXPLAIN 3
© Houghton Mifflin Harcourt Publishing Company
Next, find the natural logarithm of 13. Round
the result to the thousandths place and raise
e to that number to confirm that the power is
close to 13.
First, find the common logarithm of 13. Round
the result to the thousandths place and raise
10 to that number to confirm that the power is
close to 13.
750
Lesson 1
8/20/14 6:21 PM
Defining and Evaluating a Logarithmic Function
750
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Technology
MP.5 Discuss with students how to check their
B
0.42
First, find the common logarithm of 0.42. Round the result to the thousandths place and raise 10 to that
number to confirm that the power is close to 0.42.
log 0.42 ≈ -0.377
-0.377
≈ 0.42
10
answers for reasonableness. Reinforce that when they
are finding a common log, they are finding the power
of 10 that produces the number, and that when they
are finding the natural log, they are finding the power
of a base that is a bit less than 3.
Next, find the natural logarithm of 0.42. Round the result to the thousandths place and raise e to that
number to confirm that the power is close to 0.42.
ln 0.42 ≈ -0.868
-0.868
e
≈ 0.42
Reflect
10. For any x > 1, why is log x < ln x?
Because the base of log x is 10, the base of ln x is e, and 10 > e, the number to which 10 is
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Reasoning
MP.2 Ask students to discuss how they can tell
raised to obtain x, should be less than the number to which e is raised to obtain x.
Your Turn
Use a scientific calculator to find the common logarithm and the natural logarithm
of the given number. Verify each result by evaluating the appropriate exponential
expression.
which is greater, ln 20 or log 20, without using a
calculator. Students should reason that since ln 20
has a base of e and log 20 has a base of 10, and
because e < 10, the power of e that equals 20 must
be greater than the power of 10 that equals 20.
Therefore ln 20 > log 20.
11. 0.25
log 0.25 ≈ -0.602
10 -0.602 ≈ 0.25
© Houghton Mifflin Harcourt Publishing Company
e
-1.386
In 4 ≈ 1.386
≈ 0.25
Explain 4
e 1.386 ≈ 4
Evaluating a Logarithmic Model
There are standard scientific formulas that involve logarithms, such as the formulas for the acidity level (pH) of a
liquid and the intensity level of a sound. It’s also possible to develop your own models involving logarithms by finding
the inverses of exponential growth and decay models.
A2_MNLESE385900_U6M15L1.indd 751
Lesson 15.1
log 4 ≈ 0.602
10 0.602 ≈ 4
In 0.25 ≈ -1.386
Module 15
751
12. 4
751
Lesson 1
8/20/14 6:24 PM
Example 4

EXPLAIN 4
The acidity level, or pH, of a liquid is given by the
1
formula pH = log ____
where ⎡⎣H +⎤⎦ is the concentration
⎡ +⎤
⎣H ⎦
Evaluating a Logarithmic Model
(in moles per liter) of hydrogen ions in the liquid. In a
typical chlorinated swimming pool, the concentration of
hydrogen ions ranges from 1.58 × 10 -8 moles per liter to
6.31 × 10 -8 moles per liter. What is the range of the pH
for a typical swimming pool?
QUESTIONING STRATEGIES
When rewriting an exponential model in
logarithmic form, how do you know what
number to use for the base of the logarithmic
expression? The base will be the base from the
power in the exponential expression.
Using the pH formula, substitute the given values of ⎡⎣H +⎤⎦.
(
)
(
)
1
pH = log _
6.31 × 10 -8
≈ log 15,800,000
≈ 7.2
1
pH = log _
1.58 × 10 -8
≈ log 63,300,000
≈ 7.8
So, the pH of a swimming pool ranges from 7.2 to 7.8.

© Houghton Mifflin Harcourt Publishing Company ∙ Image Credits: (t)©ZUMA
Press, Inc./Alamy; (b)©Jupiterimages/Getty Images
Lactobacillus acidophilus is one of the bacteria used to turn milk into
yogurt. The population P of a colony of 3500 bacteria at time
t (in
t
_
minutes) can be modeled by the function P(t) = 3500(2) 73 . How long
does it take the population to reach 1,792,000?
t
_
Step 1 Solve P = 3500(2) 73 for t.
t
_
P = 3500(2) 73
Write the model.
t
_
P
_
= (2) 73
Divide both sides by 3500.
3500
Rewrite in logarithmic form.
Multiply both sides by 73.
t
P
=_
log 2 _
73
3500
P
73 log 2 _
=t
3500
Step 2 Use the logarithmic model to find t when P = 1,792,000.
1,792,000
P
t = 73 log 2 _
= 73 log 2 _
3500
3500
( ) = 657
= 73 log 2 512 = 73 9
So, the bacteria population will reach 1,792,000 in 657 minutes, or about 11 hours.
Module 15
A2_MNLESE385900_U6M15L1.indd 752
752
Lesson 1
3/24/14 2:05 PM
Defining and Evaluating a Logarithmic Function
752
Your Turn
ELABORATE
13. The intensity level L (in decibels, dB) of a sound is given by
I
the formula L = 10 log __
where I is the intensity (in watts
I
0
per square meter, W/m 2) of the sound and I 0 is the intensity
of the softest audible sound, about 10 -12 W/m 2. What is the
intensity level of a rock concert if the sound has an intensity
of 3.2 W/m 2?
QUESTIONING STRATEGIES
What is the inverse of the function
f (x) = log x? f (x) = 10 x
3.2
L = 10 log ____
-12
10
= 10 log(3.2 × 10 12)
What is the inverse of the function
f (x) = ln x? f (x) = e x
≈ 10(12.5)
= 125
So, the sound of the rock concert has an intensity level of about 125 dB.
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Critical Thinking
MP.3 Ask students to give examples of logarithmic
14. The mass (in milligrams) of beryllium-11, a radioactive isotope, in a 500-milligram sample at time t
(in seconds) is given by the function m(t) = 500e -0.05t. When will there be 90 milligrams of
beryllium-11 remaining?
Solve m = 500e -0.05t for t.
m = 500e
expressions that can be simplified using mental math,
and those that would require a calculator. Then ask
them to describe the differences between the two
types of expressions.
What is a logarithmic function? A
logarithmic function is the inverse of an
exponential function. It pairs inputs with the power
of the given base that produces the input value.
m
t = -20 In ___
500
90
= -20 In ___
500
m
___
= e -0.05t
500
m
In ___
= -0.05t
500
m
-20 In___
=t
= -20 In 0.18
≈ -20(-1.715)
500
≈ 34
© Houghton Mifflin Harcourt Publishing Company • ©Goran Djukanovic/
Shutterstock
SUMMARIZE THE LESSON
Use the model to find t when m = 90.
-0.05t
So, there will be 90 milligrams of beryllium-11 left after about 34 seconds.
Elaborate
15. What is a logarithmic function? Give an example.
A logarithmic function is a function with a given base that accepts a power of the base as
an input and outputs the exponent for the base. For example, f(x) = log 2 x is a logarithmic
function with base 2, and f(64) = log 2 64 = 6 because 64 = 2 6.
16. How can you turn an exponential model that gives y as a function of x into a logarithmic model that gives
x as a function of y?
Solve for x in terms of y in three basic steps:
1. Isolate the power of the base on one side of the equation.
2. Rewrite the equation in logarithmic form.
3. Isolate x.
17. Essential Question Check-In Write the inverse of the exponential function ƒ(x) = b x where b > 0
and b ≠ 1.
The inverse function is f -1(x) = log b x.
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753
Lesson 15.1
753
Lesson 1
8/20/14 6:27 PM
EVALUATE
Evaluate: Homework and Practice
1.
Complete the input-output table for ƒ(x) = log 2 x. Plot and label the ordered pairs
from the table. Then draw the complete graph of ƒ(x).
f(x)
4
-2
2
0.5
-1
0
1
0
2
1
4
2
8
3
x
0.25
2.
-2
y
(8, 3)
(4, 2)
(2, 1)
(1, 0)
2
4
(0.5, -1)
(0.25, -2)
• Online Homework
• Hints and Help
• Extra Practice
6
x
ASSIGNMENT GUIDE
8
-4
Use the graph of f (x) = log 2 x to do the following.
a. State the function’s domain and range using set notation.
b. Identify the function’s end behavior.
Identify the graph’s x- and y-intercepts.
c.
d. Identify the graph’s asymptotes.
e. Identify the intervals where the function has positive values and where it has
negative values.
Identify the intervals where the function is increasing and where it is decreasing
f.
c.
The graph’s only x–intercept is 1. The graph has no y-intercepts.
d.
The graph has the y–axis as an asymptote.
e.
The function has negative values on the interval (0, 1) and positive values on the
© Houghton Mifflin Harcourt Publishing Company
b.
⎫
⎧
⎧
⎫
The domain is ⎨x|x > 0⎬, and the range is ⎨y|-∞ < y < +∞⎬.
⎩
⎭
⎩
⎭
As x → +∞, f(x) → +∞. As x → 0 +, f(x) → -∞.
a.
interval (1, + ∞).
f.
The function is increasing throughout its domain. The function never decreases.
Module 15
754
Exercise
A2_MNLESE385900_U6M15L1.indd 754
Depth of Knowledge (D.O.K.)
Practice
Explore
Understanding Logarithmic
Functions as Inverses of Exponential
Functions
Exercises 1–2
Example 1
Converting Between Exponential
and Logarithmic Forms of Equations
Exercises 3–12
Example 2
Evaluating Logarithmic Functions by
Thinking in Terms of Exponents
Exercises 13–15
Example 3
Evaluating Logarithmic Functions
Using a Scientific Calculator
Exercises 16–19
Example 4
Evaluating a Logarithmic Model
Exercises 20–25
QUESTIONING STRATEGIES
How do the domain and range of f (x) = log b x
differ from the domain and range of f (x) = b x
when b > 1? The domain of f (x) = log b x is all
positive real numbers, while the domain of f (x) = b x
is all real numbers. The range of f (x) = log b x is all
real non-negative numbers, while the range of
f (x) = b x is positive numbers.
Lesson 1
Mathematical Practices
1–2
1 Recall of Information
MP.6 Precision
3–4
2 Skills/Concepts
MP.2 Reasoning
5–12
1 Recall of Information
MP.2 Reasoning
13–15
2 Skills/Concepts
MP.2 Reasoning
16–19
1 Recall of Information
MP.5 Using Tools
20–21
2 Skills/Concepts
MP.4 Modeling
1 Recall of Information
MP.4 Modeling
22
Concepts and Skills
3/25/14 4:35 AM
Defining and Evaluating a Logarithmic Function
754
Consider the exponential function ƒ(x) = 3 x.
3.
AVOID COMMON ERRORS
a. State the function’s domain and range using set notation.
b. Describe any restriction you must place on the domain of the function so that its inverse is also a
function.
When evaluating logarithmic functions for radicals,
students may think that they need to first simplify the
radical. Explain that often it is easier to write the
radical in exponential form and leave it in that form
so that its base can be compared to the base of the
logarithm.
Write the rule for the inverse function.
c.
d. State the inverse function’s domain and range using set notation.
⎧
⎧
⎫
⎫
a. The domain of f(x) is ⎨x|-∞ < x < +∞⎬, and the range is ⎨y|y > 0⎬.
⎩
⎩
⎭
⎭
b. The exponential function is one-to-one, so no restriction on its domain
is needed in order for its inverse to be a function.
c. f -1(x) = log 3 x
4.
⎫
⎫
⎧
⎧
d. The domain of f -1(x) is ⎨x|x > 0⎬, and the range is ⎨y|-∞ < y < +∞⎬.
⎭
⎭
⎩
⎩
Consider the logarithmic function ƒ(x) = log 4 x.
a. State the function’s domain and range using set notation.
b. Describe any restriction you must place on the domain of the function so that its inverse is also a
function.
Write the rule for the inverse function.
c.
d. State the inverse function’s domain and range using set notation.
⎫
⎧
⎧
⎫
a. The domain of f (x) is ⎨x|x > 0⎬, and the range is ⎨y|-∞ < y < +∞⎬.
⎭
⎩
⎩
⎭
b. The logarithmic function is one-to-one, so no restriction on its domain
is needed in order for its inverse to be a function.
© Houghton Mifflin Harcourt Publishing Company
c. f -1(x) = 4 x
⎫
⎫
⎧
⎧
d. The domain of f -1(x) is ⎨x|-∞ < x < +∞⎬, and the range is ⎨y|y > 0⎬.
⎭
⎭
⎩
⎩
Write the given exponential equation in logarithmic form.
-2
1
_
5. 5 3 = 125
6.
= 100
10
log 5 125 = 3
log __1 100 = -2
( )
3 =n
m
7.
8.
log 3 n = m
log _1 q = p
2
Write the given logarithmic equation in exponential form.
log 6 1296 = 4
9.
6 = 1296
4
11. log 8 x = y
1 =3
10. log _1 _
4 64
1
(_41 ) = _
64
3
12. log _2 c = d
3
(_32 ) = c
8 =x
y
Module 15
Exercise
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23
24–25
755
Lesson 15.1
()
10
p
1 =q
_
2
d
755
Depth of Knowledge (D.O.K.)
Lesson 1
Mathematical Practices
2 Skills/Concepts
MP.3 Logic
3 Strategic Thinking
MP.3 Logic
8/20/14 6:30 PM
( )
1 , and ƒ (√_
13. If ƒ (x) = log 3 x, find ƒ (243), ƒ _
27 ).
27
f(243) = y
f
log 3 243 = y
1
=y
(_
27 )
log 3
3 y = 243
1
_
=y
27
1
3 =_
27
1
3 =_
―
log √―
27 = y
3
y
y=5
33
3 y = 3 -3
So, f (243) = 5.
y = -3
1
So, f(_) = -3.
27
()
3 ―
1 and ƒ (6 √
14. If ƒ(x) = log 6 x, find ƒ (36), ƒ _
6)
6
1
f (36) = y
=y
f
6
1
log 6 36 = y
log 6 = y
6
y
1
6 = 36
6y =
6
6y = 62
6 y = 6 -1
(_)
_
y=2
_
y = -1
So, f (36) = 2.
So, f
(_16 ) = -1.
f
1
=y
(_
64 )
log _1
3 = 32
3
y=
2
3
So, f ( √27 ) = .
2
4
― _
―
3
6) = y
f(6
―
3
log 6(6
6) = y
6y =
6 =
y
4
(_1 )
64
y
4
(_1 )
y
()
1
= _
4
4
y=3
1
So, f
= 3.
64
(_)
Module 15
A2_MNLESE385900_U6M15L1 756
(_14 ) = 256
(_14 ) = 4
(_14 ) = (_41 )
_
_
y
16
(_14 ) = ―
(_14 ) = ―4
(_14 ) = 4_
_
(_14 ) = (_14 )
y
y
4
3
―
6
_4
―
y
1
=_
43
6 ∙6
3
4

6y = 6 3
4
y=
3
4
3 ―
6) = .
So, f(6
3
4
y
y
―
3 ―
3

3
6 y = 6
6
3
log _1 
16 = y
4
(_1 ) = _1
_
―
log _1 256 = y
64
3
_3
y
3
f (
16 ) = y
f (256) = y
1
_
=y
―
―
= √3
© Houghton Mifflin Harcourt Publishing Company
( )
3y
3 ―
1
15. If ƒ (x) = log _1 x, find ƒ __
, ƒ (256), and ƒ( √16 )
64
4
To help students reinforce their understanding of
common and natural logarithms, encourage them to
try to approximate their answers before entering the
expressions into a calculator.
3 y = √27
y
3y = 35
TECHNOLOGY
f ( √27 ) = y
-4
y = -4
So, f (256) = -4.
756
y
3

3
2

2
3
y
-2
3
_
_
y = -2
3
3 ―
So, f
16 = - 2
3
Lesson 1
10/16/14 10:31 AM
Defining and Evaluating a Logarithmic Function
756
Use a scientific calculator to find the common logarithm and
the natural logarithm of the given number. Verify each result by
evaluating the appropriate exponential expression.
KINESTHETIC EXPERIENCE
To help students remember how to convert from
logarithmic form to exponential form, have them
repeatedly trace a counterclockwise “circle” in the
equation using their fingers. For example, in the
equation log b a = x, they should trace the path from
b to x to a, while saying “b to the x equals a.”
Encourage students to do this when rewriting
numerical examples, as well.
16. 19
17. 9
log 9 ≈ 0.954
log 19 ≈ 1.279
10
1.279
≈ 19
10 0.954 ≈ 9
In 19 ≈ 2.944
e
2.944
In 9 ≈ 2.197
≈ 19
e 2.197 ≈ 9
18. 0.6
19. 0.31
log 0.31 ≈ -0.509
log 0.6 ≈ -0.222
10 -0.222 ≈ 0.6
10 -0.509 ≈ 0.31
In 0.6 ≈ -0.511
e
-0.511
In 0.31 ≈ -1.171
≈ 0.6
e -1.171 ≈ 0.31
20. The acidity level, or pH, of a liquid is given by the formula
pH = log 1+ where ⎡⎣H +⎤⎦ is the concentration (in moles
⎡⎣H ⎤⎦
per liter) of hydrogen ions in the liquid. What is the pH of iced
tea with a hydrogen ion concentration of 0.000158 mole per liter?
_
© Houghton Mifflin Harcourt Publishing Company • (t)©pilipphoto/
Shutterstock; (b)©Christian Delbert/Shutterstock
pH = log
1
_
0.000158
≈ log(6329)
≈ 3.8
The pH of the iced tea is about 3.8.
21. The intensity level L (in decibels, dB) of a sound is
I
given by the formula L = 10 log __
where I is the
I0
intensity (in watts per square meter, W/m 2) of the sound
and I 0 is the intensity of the softest audible sound, about
10 -12 W/m 2. What is the intensity level of a lawn mower
if the sound has an intensity of 0.00063 W/m 2?
L = 10 log
0.00063
_
10 -12
=10 log(6.3 × 10 8)
≈ 10(8.8)
≈ 88
So, the sound of the lawn mower has an intensity
level of about 88 dB.
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Lesson 15.1
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Lesson 1
3/24/14 2:03 PM
22. Match each liquid with its pH given the concentration of hydrogen ions in the liquid.
Liquid
Hydrogen Ion
Concentration
CONNECT VOCABULARY
pH
A. Cocoa
5.2 × 10
-7
G 3.5
B. Cider
7.9 × 10 -4
C 3.3
C. Ginger Ale
4.9 × 10 -4
F 2.4
D. Honey
1.3 × 10 -4
E 4.5
E. Buttermilk
3.2 × 10 -5
A 6.3
F. Cranberry juice
4.0 × 10 -3
I
G. Pinneapple juice
3.1 × 10 -4
B 3.1
H. Tomato juice
6.3 × 10 -2
H 1.2
I. Carrot juice
4.0 × 10 -7
D 3.9
To tap into prior knowledge, have students articulate
how logarithmic functions are related to exponential
functions (inverses) and how they can use the rules
of exponents with logarithms.
6.4
H.O.T. Focus on Higher Order Thinking
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©lrlucik/
Shutterstock
23. Explain the Error Jade is taking a chemistry test and has to find the pH of a liquid
given that its hydrogen ion concentration is 7.53 × 10 -9 moles per liter. She writes
the following.
1
pH = ln _
[H +]
1
= ln _
7.53 × 10 –9
≈ 18.7
She knows that the pH scale ranges from 1 to 14, so her answer of 18.7 must be
incorrect, but she runs out of time on the test. Explain her error and find the
correct pH.
The formula for finding the pH uses the common logarithm, not the
natural logarithm. She should have found
1
log ___
+
⎡⎣H ⎤⎦
1
pH = log ___
+
⎡⎣H ⎤⎦
1
= log _________
-9
≈ 8.1
7.53 ×10
So, the pH of the liquid is about 8.1.
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Lesson 1
21/08/14 4:23 PM
Defining and Evaluating a Logarithmic Function
758
24. Multi-step Exponential functions have the general form ƒ(x) = ab x - h + k where
a, b, h, and k are constants, a ≠ 0, b > 0 , and b ≠ 1.
PEER-TO-PEER DISCUSSION
a. State the domain and range of ƒ(x) using set notation.
Ask students to discuss with a partner how to find
the value of ln __1e without using a calculator. Then have
them discuss their reasoning with the class.
1
1
1
If ln e = x, then e x = e . Since e = e -1, x = -1.
_
_
b. Show how to find ƒ -1(x). Give a description of each step you take.
State the domain and range of ƒ -1(x) using set notation.
⎧
⎫
⎧
⎫
a. The domain of f(x) is ⎨x|-∞ < x < +∞⎬, and the range is ⎨y|y > k⎬ if a > 0
⎩
⎭
⎩
⎭
⎧
⎫
or ⎨y|y < k⎬ if a < 0.
⎩
⎭
b.
c.
_
f(x) = ab x - h + k
JOURNAL
Have students define a logarithmic function and
describe how to transform an exponential function
into a logarithmic function.
y = ab
x-h
y - k = ab
x-h
y____
-k
x-h
a =b
y____
-k
log b a = x - h
y-k
h + log b ____
a =x
x____
-k
h + log b a = y
x-k
-1
h + log b ____
a = f (x)
+k
Write the function.
Replace f(x) with y.
Subtract k from both sides.
Divide both sides by a.
Rewrite the equation in logarithmic form.
Add h to both sides.
Switch x and y.
Replace y with f -1(x)
⎧
⎧
⎫
⎫
The domain of f -1(x) is ⎨x|x > k⎬ if a > 0 or ⎨x|x < k⎬ if a < 0, and the range
⎩
⎩
⎭
⎭
⎧
⎫
is ⎨y|-∞ < y < +∞⎬.
⎩
⎭
c.
25. Justify Reasoning Evaluate each expression without using a calculator. Explain
your reasoning.
a. ln e 2
b. 10 log 7
4 log
© Houghton Mifflin Harcourt Publishing Company
c.
2
5
Let f(x) = ln x. Then f -1(x) = e x.
a.
ln e 2 = f(e 2)
c. 4 log 5 = (2 2)
2
= 2 2log
= f(f -1 (2))
=2
Let f(x) = 10 x. Then f -1(x) = log x.
b.
10
log7
= f(log7)
= f(f -1 (7))
= 2 log
2
log 2 5
2
5
5 + log 2 5
= (2 log
2
5
)(2 log 5)
2
Let g(x) = 2 . Then g (x) = log 2 x.
-1
x
2 log 5 = g(log 2 5)
2
= g(g -1(5))
=5
=7
Substitute 5 for 2 log 5 in 4 log 5 = (2 log
2
4
log 2 5
= (2
log 2 5
)(2
= (5)(5)
log 2 5
)
2
2
5
)(2 log 5)
2
= 25
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Lesson 15.1
759
Lesson 1
8/20/14 6:33 PM
Lesson Performance Task
AVOID COMMON ERRORS
Skydivers use an instrument called an altimeter to determine their height above Earth’s
surface. An altimeter measures atmospheric pressure and converts it to altitude based on
the relationship between pressure and altitude. One model for atmospheric pressure
P (in kilopascals, kPa) as a function of altitude a (in kilometers) is P = 100e -a/8.
Students may solve for a by taking the common
logarithm of both sides of the equation rather than
the natural logarithm, ln. Explain that the natural log
has base e, and because the equation has e raised to a
power, they must take the natural log in order to
isolate a.
a. Since an altimeter measures pressure directly, pressure is the independent variable for
an altimeter. Rewrite the model P = 100e -a/8 so that it gives altitude as a function of
pressure.
b. To check the function in part a, use the fact that atmospheric pressure at Earth’s
surface is about 100 kPa.
c. Suppose a skydiver deploys the parachute when the altimeter measures 87 kPa. Use
the function in part a to determine the skydiver’s altitude. Give your answer in both
kilometers and feet. (1 kilometer ≈ 3281 feet)
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Math Connections
MP.1 Have students discuss the significance of the
Solve P = 100e -a/8 for a.
a.
P = 100e -a/8
P
___
= e -a/8
100
P
a
ln ___
= -_
100
8
P
-8 ln ___
=a
negative sign in the formula for P. Have students
explain why the formula for a has a negative sign and
yet the altitude must be a positive number.
100
b.
100
P
Substitution 100 for P in a = -8 ln___
gives a = -8 ln ___
= -8 ln 1 = -8(0) = 0.
100
100
Since an altitude of 0 km corresponds to being on Earth’s surface, the function checks.
c.
87
P
Substitution 87 for P in a = -8 ln___
gives a = -8 ln___
= -8 ln 0.87 ≈ 1.1.
100
100
So, the skydiver opens the parachute at an altitude of about 1.1 kilometers,
or 1.1(3281) ≈ 3600 feet.
© Houghton Mifflin Harcourt Publishing Company
Module 15
760
Lesson 1
EXTENSION ACTIVITY
A2_MNLESE385900_U6M15L1.indd 760
Have students research the height at which pilots are required to wear a pressure
suit to protect them from low atmospheric pressure. about 15,000 meters Have
students calculate the atmospheric pressure at this altitude, and then have them
calculate the atmospheric pressure at the top of Mount Everest. 15 kPa and
33 kPa Have students discuss whether mountain climbers are in danger from the
effects of low atmospheric pressure.
8/20/14 6:35 PM
Scoring Rubric
2 points: Student correctly solves the problem and explains his/her reasoning.
1 point: Student shows good understanding of the problem but does not fully
solve or explain his/her reasoning.
0 points: Student does not demonstrate understanding of the problem.
Defining and Evaluating a Logarithmic Function
760