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Section 3.3 Properties of Logarithms
SECTION 3.3
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Objectives
Use the product rule.
Use the quotient rule.
Use the power rule.
Expand logarithmic
expressions.
Condense logarithmic
expressions.
Use the change-of-base
property.
441
Properties of Logarithms
We all learn new things in different ways. In this
section, we consider important properties of
logarithms. What would be the most effective
way for you to learn these properties?
Would it be helpful to use your graphing
utility and discover one of these properties
for yourself? To do so, work Exercise 133
in Exercise Set 3.2 before continuing. Would
it be helpful to evaluate certain logarithmic
expressions that suggest three of the
properties? If this is the case, work Preview
Exercises 147–149 in Exercise Set 3.2
before continuing. Would the properties
become more meaningful if you could see exactly where they come from? If so,
you will find details of the proofs of many of these properties in the appendix. The
remainder of our work in this chapter will be based on the properties of logarithms
that you learn in this section.
The Product Rule
Use the product rule.
Properties of exponents correspond to properties of logarithms. For example, when
we multiply with the same base, we add exponents:
DISCOVERY
We know that log 100,000 = 5.
Show that you get the same result
by writing 100,000 as 1000 # 100
and then using the product rule.
Then verify the product rule
by using other numbers whose
logarithms are easy to find.
bm # b n = b m + n .
This property of exponents, coupled with an awareness that a logarithm is an
exponent, suggests the following property, called the product rule:
The Product Rule
You can find the
proof of the
product rule in
the appendix.
Let b, M, and N be positive real numbers with b ⬆ 1.
log b(MN) = log b M + log b N
The logarithm of a product is the sum of the logarithms.
When we use the product rule to write a single logarithm as the sum of two
logarithms, we say that we are expanding a logarithmic expression. For example, we
can use the product rule to expand ln(7x):
ln(7x) = ln 7 + ln x.
The logarithm
of a product
EXAMPLE 1
is
the sum of
the logarithms.
Using the Product Rule
Use the product rule to expand each logarithmic expression:
a. log4(7 # 5)
b. log(10x).
442
Chapter 3 Exponential and Logarithmic Functions
SOLUTION
a. log 4(7 # 5) = log 4 7 + log 4 5 The logarithm of a product is the sum of the
logarithms.
b. log(10x) = log 10 + log x
The logarithm of a product is the sum of the
logarithms. These are common logarithms with
base 10 understood.
= 1 + log x
Because logb b = 1, then log 10 = 1.
● ● ●
Check Point 1 Use the product rule to expand each logarithmic expression:
a. log6(7 # 11)
�
Use the quotient rule.
b. log(100x).
The Quotient Rule
When we divide with the same base, we subtract exponents:
bm
= b m - n.
bn
This property suggests the following property of logarithms, called the quotient rule:
DISCOVERY
We know that log 2 16 = 4. Show
that you get the same result by
32
and then using
writing 16 as
2
the quotient rule. Then verify the
quotient rule using other numbers
whose logarithms are easy to find.
The Quotient Rule
Let b, M, and N be positive real numbers with b ⬆ 1.
M
b = log b M - log b N
N
The logarithm of a quotient is the difference of the logarithms.
log b a
When we use the quotient rule to write a single logarithm as the difference of two
logarithms, we say that we are expanding a logarithmic expression. For example, we
x
can use the quotient rule to expand log :
2
x
log a b = log x - log 2.
2
The logarithm
of a quotient
EXAMPLE 2
is
the difference of
the logarithms.
Using the Quotient Rule
Use the quotient rule to expand each logarithmic expression:
19
e3
b. ln ¢ ≤.
a. log7 a b
x
7
SOLUTION
a. log 7 a
b. ln ¢
19
b = log 7 19 - log 7 x
x
e3
≤ = ln e 3 - ln 7
7
= 3 - ln 7
The logarithm of a quotient is the difference of
the logarithms.
The logarithm of a quotient is the difference of
the logarithms. These are natural logarithms
with base e understood.
Because ln ex = x, then ln e3 = 3.
● ● ●
Section 3.3 Properties of Logarithms
443
Check Point 2 Use the quotient rule to expand each logarithmic expression:
a. log8 a
�
Use the power rule.
23
b
x
b. ln ¢
e5
≤.
11
The Power Rule
When an exponential expression is raised to a power, we multiply exponents:
n
(bm) = bmn.
This property suggests the following property of logarithms, called the power rule:
The Power Rule
Let b and M be positive real numbers with b ⬆ 1, and let p be any real number.
log b M p = p log b M
The logarithm of a number with an exponent is the product of the exponent and
the logarithm of that number.
When we use the power rule to “pull the exponent to the front,” we say that we
are expanding a logarithmic expression. For example, we can use the power rule to
expand ln x2:
ln x2 = 2 ln x.
The logarithm of
a number with an
exponent
is
the product of the
exponent and the
logarithm of that number.
Figure 3.18 shows the graphs of y = ln x2 and y = 2 ln x in [-5, 5, 1] by [-5, 5, 1]
viewing rectangles. Are ln x2 and 2 ln x the same? The graphs illustrate that y = ln x2
and y = 2 ln x have different domains. The graphs are only the same if x 7 0. Thus,
we should write
ln x2 = 2 ln x for x 7 0.
y = 2 ln x
y = ln x2
Domain: (− , 0) ´ (0, )
Domain: (0, )
FIGURE 3.18 ln x2 and 2 ln x have different domains.
When expanding a logarithmic expression, you might want to determine whether
the rewriting has changed the domain of the expression. For the rest of this section,
assume that all variables and variable expressions represent positive numbers.
EXAMPLE 3
Using the Power Rule
Use the power rule to expand each logarithmic expression:
a. log 5 74
b. ln 1x
c. log(4x)5.
444
Chapter 3 Exponential and Logarithmic Functions
SOLUTION
a. log 5 74 = 4 log 5 7
The logarithm of a number with an exponent is the
exponent times the logarithm of the number.
1
b. ln 1x = ln x 2
=
1
2 ln
Rewrite the radical using a rational exponent.
x
Use the power rule to bring the exponent to the front.
c. log(4x) = 5 log(4x)
5
We immediately apply the power rule because the entire
variable expression, 4x, is raised to the 5th power.
● ● ●
Check Point 3 Use the power rule to expand each logarithmic expression:
3
b. ln 1
x
a. log 6 39
�
Expand logarithmic expressions.
c. log(x + 4)2.
Expanding Logarithmic Expressions
It is sometimes necessary to use more than one property of logarithms when you
expand a logarithmic expression. Properties for expanding logarithmic expressions
are as follows:
Properties for Expanding Logarithmic Expressions
For M 7 0 and N 7 0:
1. logb (MN) = logb M + logb N Product rule
M
2. logb a b = logb M - logb N Quotient rule
N
3. log b M p = p log b M
Power rule
GREAT QUESTION!
Are there some common screw-ups that I can avoid when using properties of logarithms?
Try to avoid the following errors:
The graphs show that
ln(x+3) ln x+ln 3.
y = ln x shifted
3 units left
y = ln x shifted
ln 3 units up
In general,
log b(M + N) ⬆ log b M + log b N.
Incorrect!
logb (M + N) = logb M + logb N
logb (M - N) = logb M - logb N
logb (M # N) = logb M # logb N
logb a
logb M
M
b =
N
logb N
log b M
y1 = ln(x + 3)
log b N
= log b M - log b N
log b (MN p) = p log b (MN)
y2 = ln x + ln 3
[−4, 5, 1] by [−3, 3, 1]
EXAMPLE 4
Expanding Logarithmic Expressions
Use logarithmic properties to expand each expression as much as possible:
3
1
x
a. logb(x2 1y)
b. log 6 ¢
≤.
36y 4
Section 3.3 Properties of Logarithms
445
SOLUTION
We will have to use two or more of the properties for expanding logarithms in each
part of this example.
a. logb (x2 1y) = logb 1 x2 y 2 2
1
Use exponential notation.
1
2
= logb x2 + logb y
1
= 2 log b x + log b y
2
Use the product rule.
Use the power rule.
1
3
1
x
x3
b. log 6 ¢
≤
=
log
¢
≤
6
36y 4
36y 4
Use exponential notation.
1
= log6 x 3 - log6 (36y4)
Use the quotient rule.
1
3
= log6 x - (log6 36 + log6 y4)
1
log6 x - (log6 36 + 4 log6 y)
3
1
= log6 x - log6 36 - 4 log6 y
3
1
= log 6 x - 2 - 4 log 6 y
3
=
Use the product rule on
log6 (36y4).
Use the power rule.
Apply the distributive property.
log6 36 = 2 because 2 is the
power to which we must raise
6 to get 36. (62 = 36)
● ● ●
Check Point 4 Use logarithmic properties to expand each expression as much
as possible:
3
a. log b 1 x4 1
y2
�
Condense logarithmic
expressions.
b. log5 ¢
1x
≤.
25y3
Condensing Logarithmic Expressions
To condense a logarithmic expression, we write the sum or difference of two or more
logarithmic expressions as a single logarithmic expression. We use the properties of
logarithms to do so.
GREAT QUESTION!
Properties for Condensing Logarithmic Expressions
Are the properties listed on the
right the same as those in the box
on page 444?
For M 7 0 and N 7 0:
Yes. The only difference is that
we’ve reversed the sides in each
property from the previous box.
1. logb M + logb N = logb (MN) Product rule
M
2. logb M - logb N = logb a b Quotient rule
N
3. p logb M = logb Mp
Power rule
EXAMPLE 5
Condensing Logarithmic Expressions
Write as a single logarithm:
a. log4 2 + log4 32
b. log(4x - 3) - log x.
SOLUTION
a. log 4 2 + log 4 32 = log 4(2 # 32)
= log 4 64
= 3
b. log(4x - 3) - log x = loga
Use the product rule.
We now have a single logarithm.
However, we can simplify.
log4 64 = 3 because 43 = 64.
4x - 3
b
x
Use the quotient rule.
● ● ●
446
Chapter 3 Exponential and Logarithmic Functions
Check Point 5 Write as a single logarithm:
a. log 25 + log 4
b. log (7x + 6) - log x.
Coefficients of logarithms must be 1 before you can condense them using the
product and quotient rules. For example, to condense
2 ln x + ln(x + 1),
the coefficient of the first term must be 1. We use the power rule to rewrite the
coefficient as an exponent:
1. Use the power rule to make the number in front an exponent.
2 ln x+ln(x+1)=ln x2+ln(x+1)=ln[x2(x+1)].
2. Use the product rule. The sum of logarithms with
coefficients of 1 is the logarithm of the product.
EXAMPLE 6
Condensing Logarithmic Expressions
Write as a single logarithm:
a. 12 log x + 4 log (x - 1)
c. 4 logb x - 2 logb 6 -
1
2
b. 3 ln (x + 7) - ln x
logb y.
SOLUTION
a.
1
2
log x + 4 log (x - 1)
1
= log x 2 + log (x - 1)4
Use the power rule so that all coefficients are 1.
1
2
= log 3 x (x - 1)4 4
Use the product rule. The condensed form
can be expressed as log[ 1x (x - 1)4].
b. 3 ln(x + 7) - ln x
= ln(x + 7)3 - ln x
(x + 7)3
R
= ln J
x
c. 4 logb x - 2 logb 6 -
Use the power rule so that all coefficients are 1.
Use the quotient rule.
1
2
logb y
1
= logb x4 - logb 62 - logb y 2
= logb x4 -
1 logb 36
+ logb y
1
2
= logb x4 - logb 1 36y
= logb ¢
4
x
1≤
36y 2
Use the power rule so that all coefficients are 1.
1
2
2
or logb ¢
2
Rewrite as a single subtraction.
Use the product rule.
4
x
≤
361y
Use the quotient rule.
● ● ●
Check Point 6 Write as a single logarithm:
a. 2 ln x +
1
3
ln (x + 5)
b. 2 log (x - 3) - log x
c. 14 logb x - 2 logb 5 - 10 logb y.
�
Use the change-of-base
property.
The Change-of-Base Property
We have seen that calculators give the values of both common logarithms (base 10)
and natural logarithms (base e). To find a logarithm with any other base, we can use
the following change-of-base property:
Section 3.3 Properties of Logarithms
447
The Change-of-Base Property
For any logarithmic bases a and b, and any positive number M,
log a M
.
log a b
The logarithm of M with base b is equal to the logarithm of M with any new base
divided by the logarithm of b with that new base.
log b M =
In the change-of-base property, base b is the base of the original logarithm.
Base a is a new base that we introduce. Thus, the change-of-base property allows us
to change from base b to any new base a, as long as the newly introduced base is a
positive number not equal to 1.
The change-of-base property is used to write a logarithm in terms of quantities
that can be evaluated with a calculator. Because calculators contain keys for common
(base 10) and natural (base e) logarithms, we will frequently introduce base 10 or
base e.
Change-of-Base
Property
logbM=
Introducing Common
Logarithms
logaM
logab
logbM=
a is the new
introduced base.
log10M
log10b
Introducing Natural
Logarithms
logbM=
10 is the new
introduced base.
logeM
logeb
e is the new
introduced base.
Using the notations for common logarithms and natural logarithms, we have the
following results:
The Change-of-Base Property: Introducing Common
and Natural Logarithms
Introducing Common Logarithms
log b M =
log M
log b
EXAMPLE 7
Introducing Natural Logarithms
log b M =
ln M
ln b
Changing Base to Common Logarithms
Use common logarithms to evaluate log5 140.
SOLUTION
DISCOVERY
Find a reasonable estimate of
log5 140 to the nearest whole
number. To what power can
you raise 5 in order to get 140?
Compare your estimate to the
value obtained in Example 7.
log M
,
log b
log 140
log 5 140 =
log 5
⬇ 3.07.
Use a calculator: 140 冷 LOG 冷 冷 , 冷 5 冷 LOG 冷 冷 = 冷
Because log b M =
or 冷 LOG 冷 140 冷 , 冷 冷 LOG 冷 5 冷 ENTER 冷. On some
calculators, parentheses are needed after 140 and 5.
This means that log 5 140 ⬇ 3.07.
Check Point 7 Use common logarithms to evaluate log7 2506.
● ● ●
448
Chapter 3 Exponential and Logarithmic Functions
EXAMPLE 8
Changing Base to Natural Logarithms
Use natural logarithms to evaluate log 5 140.
SOLUTION
ln M
,
ln b
ln 140
log5 140 =
ln 5
⬇ 3.07. Use a calculator: 140 冷 LN 冷 冷 , 冷 5 冷 LN 冷 冷 = 冷
Because logb M =
or 冷 LN 冷 140 冷 , 冷 冷 LN 冷 5 冷 ENTER 冷. On some calculators,
parentheses are needed after 140 and 5.
We have again shown that log 5 140 ⬇ 3.07.
● ● ●
Check Point 8 Use natural logarithms to evaluate log7 2506.
TECHNOLOGY
We can use the change-of-base property to graph
logarithmic functions with bases other than 10 or e on
a graphing utility. For example, Figure 3.19 shows the
graphs of
y = log2 x
y = log20 x
y = log2 x and y = log20 x
in a [0, 10, 1] by [-3, 3, 1] viewing rectangle. Because
ln x
ln x
log2 x =
and log20 x =
, the functions are
ln 2
ln 20
entered as
y1= LN x ÷ LN 2
FIGURE 3.19 Using the
change-of-base property to graph
logarithmic functions
and y2= LN x ÷ LN 20.
On some calculators, parentheses
are needed after x, 2, and 20.
CONCEPT AND VOCABULARY CHECK
Fill in each blank so that the resulting statement is true.
1. The product rule for logarithms states that
log b(MN) =
. The logarithm of a
product is the
of the logarithms.
2. The quotient rule for logarithms states that
M
log b a b =
. The logarithm of a
N
quotient is the
of the logarithms.
3. The power rule for logarithms states that
log b M p =
. The logarithm of a number
with an exponent is the
of the exponent
and the logarithm of that number.
4. The change-of-base property for logarithms allows
us to write logarithms with base b in terms of a new
base a. Introducing base a, the property states that
logb M =
.
Section 3.3 Properties of Logarithms
449
EXERCISE SET 3.3
Practice Exercises
In Exercises 1–40, use properties of logarithms to expand each
logarithmic expression as much as possible. Where possible,
evaluate logarithmic expressions without using a calculator.
1. log5(7 # 3)
4. log9(9x)
7
7. log7 a b
x
x
10. log a
b
1000
e2
13. ln¢ ≤
5
16. logb x7
2. log8(13 # 7)
5. log(1000x)
9
8. log9 a b
x
64
11. log4 a b
y
e4
14. ln¢ ≤
8
17. log N -6
18. log M -8
5
19. ln1
x
7
20. ln1
x
21. logb(x2 y)
22. logb(xy3)
23. log4 ¢
36
25. log6 ¢
2x + 1
x3 y
28. logb ¢ 2 ≤
z
3 x
31. log
Ay
34. logb ¢
3
1
xy4
z
5
≤ 26. log8 ¢
≤
3. log7(7x)
6. log(10,000x)
x
9. log a
b
100
125
12. log5 a
b
y
15. log b x 3
1x
≤
64
24. log5 ¢
64
2x + 1
≤ 27. logb ¢
x
Ay
33. logb ¢
x2 y
B 25
3
x 2x2 + 1
R
(x + 1)4
39. log J
3
10x2 2 1 - x
R
7(x + 1)2
x y
z
2
38. lnJ
x4 2x2 + 3
40. log J
(x + 3)5
≤
1xy3
≤
R
3
100x3 2 5 - x
R
3(x + 7)2
In Exercises 41–70, use properties of logarithms to condense each
logarithmic expression. Write the expression as a single logarithm
whose coefficient is 1. Where possible, evaluate logarithmic
expressions without using a calculator.
41.
43.
45.
47.
49.
51.
53.
55.
57.
59.
61.
log 5 + log 2
ln x + ln 7
log2 96 - log2 3
log(2x + 5) - log x
log x + 3 log y
1
2 ln x + ln y
2 logb x + 3 logb y
5 ln x - 2 ln y
3 ln x - 13ln y
4 ln(x + 6) - 3 ln x
3 ln x + 5 ln y - 6 ln z
42.
44.
46.
48.
50.
52.
54.
56.
58.
60.
62.
log 250 + log 4
ln x + ln 3
log3 405 - log3 5
log(3x + 7) - log x
log x + 7 log y
1
3 ln x + ln y
5 logb x + 6 logb y
7 ln x - 3 ln y
2 ln x - 12ln y
8 ln(x + 9) - 4 ln x
4 ln x + 7 ln y - 3 ln z
63.
1
2 (log
64.
1
3 (log4 x
65.
66.
67.
68.
1
2 (log5 x
1
3 (log4 x
1
3 [2 ln(x
1
3 [5 ln(x
x + log y)
+
+
+
log5 y) - 2 log5(x
log4 y) + 2 log4(x
5) - ln x - ln(x2
6) - ln x - ln(x2
+
+
-
1)
1)
4)]
25)]
- log4 y)
72. log6 17
75. log0.1 17
78. logp 400
73. log14 87.5
76. log0.3 19
In Exercises 79–82, use a graphing utility and the change-of-base
property to graph each function.
80. y = log15 x
82. y = log3(x - 2)
79. y = log3 x
81. y = log2(x + 2)
In Exercises 83–88, let logb 2 = A and logb 3 = C. Write each
expression in terms of A and C.
1x
≤
25
z3
xy 4
36. log 2 5
B 16
3
37. lnJ
71. log5 13
74. log16 57.2
77. logp 63
Practice Plus
30. ln1ex
5
35. log 5
In Exercises 71–78, use common logarithms or natural logarithms
and a calculator to evaluate to four decimal places.
83. logb 32
84. logb 6
85. logb 8
86. logb 81
2
87. logb
A 27
3
88. logb
A 16
2
29. log 2100x
32. log
69. log x + log(x2 - 1) - log 7 - log(x + 1)
70. log x + log(x2 - 4) - log 15 - log(x + 2)
In Exercises 89–102, determine whether each equation is true or
false. Where possible, show work to support your conclusion. If
the statement is false, make the necessary change(s) to produce a
true statement.
89.
91.
93.
95.
ln e = 0
log4(2x3) = 3 log4(2x)
x log 10x = x 2
ln(5x) + ln 1 = ln(5x)
97. log(x + 3) - log(2x) =
90.
92.
94.
96.
log(x +
ln 0 = e
ln(8x3) = 3 ln(2x)
ln(x + 1) = ln x + ln 1
ln x + ln(2x) = ln(3x)
3)
log(2x)
log(x + 2)
= log(x + 2) - log(x - 1)
log(x - 1)
x - 1
99. log6 ¢ 2
≤ = log6(x - 1) - log6(x2 + 4)
x + 4
100. log6[4(x + 1)] = log6 4 + log6(x + 1)
1
1
101. log3 7 =
102. ex =
log7 3
ln x
98.
Application Exercises
103. The loudness level of a sound can be expressed by
comparing the sound’s intensity to the intensity of a sound
barely audible to the human ear. The formula
D = 10(log I - log I0)
describes the loudness level of a sound, D, in decibels, where
I is the intensity of the sound, in watts per meter2, and I0 is
the intensity of a sound barely audible to the human ear.
a. Express the formula so that the expression in parentheses
is written as a single logarithm.
b. Use the form of the formula from part (a) to answer this
question: If a sound has an intensity 100 times the intensity
of a softer sound, how much larger on the decibel scale is
the loudness level of the more intense sound?
450
Chapter 3 Exponential and Logarithmic Functions
104. The formula
1
t = [ln A - ln(A - N)]
c
describes the time, t, in weeks, that it takes to achieve
mastery of a portion of a task, where A is the maximum
learning possible, N is the portion of the learning that
is to be achieved, and c is a constant used to measure an
individual’s learning style.
a. Express the formula so that the expression in brackets is
written as a single logarithm.
b. The formula is also used to determine how long it will
take chimpanzees and apes to master a task. For example,
a typical chimpanzee learning sign language can master a
maximum of 65 signs. Use the form of the formula from
part (a) to answer this question: How many weeks will it
take a chimpanzee to master 30 signs if c for that chimp
is 0.03?
Writing in Mathematics
105. Describe the product rule for logarithms and give an
example.
106. Describe the quotient rule for logarithms and give an
example.
107. Describe the power rule for logarithms and give an example.
108. Without showing the details, explain how to condense
ln x - 2 ln(x + 1).
109. Describe the change-of-base property and give an example.
110. Explain how to use your calculator to find log 14 283.
111. You overhear a student talking about a property of
logarithms in which division becomes subtraction. Explain
what the student means by this.
112. Find ln 2 using a calculator. Then calculate each of the
following: 1 - 12;
1 - 12 + 13;
1 - 12 + 13 - 14;
1
1
1
1
1 - 2 + 3 - 4 + 5; . . . . Describe what you observe.
Technology Exercises
113. a. Use a graphing utility (and the change-of-base property)
to graph y = log 3 x.
b. Graph y = 2 + log3 x, y = log3(x + 2), and y = -log3 x
in the same viewing rectangle as y = log3 x. Then
describe the change or changes that need to be made
to the graph of y = log 3 x to obtain each of these three
graphs.
114. Graph y = log x, y = log(10x), and y = log(0.1x) in the
same viewing rectangle. Describe the relationship among
the three graphs. What logarithmic property accounts for
this relationship?
115. Use a graphing utility and the change-of-base property to
graph y = log3 x, y = log25 x, and y = log 100 x in the same
viewing rectangle.
a. Which graph is on the top in the interval (0, 1)? Which is
on the bottom?
b. Which graph is on the top in the interval (1, ⬁ )? Which
is on the bottom?
c. Generalize by writing a statement about which graph is
on top, which is on the bottom, and in which intervals,
using y = log b x where b 7 1.
Disprove each statement in Exercises 116–120 by
a. letting y equal a positive constant of your choice, and
b. using a graphing utility to graph the function on each side
of the equal sign. The two functions should have different
graphs, showing that the equation is not true in general.
116. log(x + y) = log x + log y
log x
x
117. log a b =
118. ln(x - y) = ln x - ln y
y
log y
ln x
119. ln(xy) = (ln x)(ln y)
120.
= ln x - ln y
ln y
Critical Thinking Exercises
Make Sense? In Exercises 121–124, determine whether each
statement makes sense or does not make sense, and explain your
reasoning.
121. Because I cannot simplify the expression bm + bn by adding
exponents, there is no property for the logarithm of a sum.
122. Because logarithms are exponents, the product, quotient,
and power rules remind me of properties for operations
with exponents.
123. I can use any positive number other than 1 in the changeof-base property, but the only practical bases are 10 and e
because my calculator gives logarithms for these two bases.
x
124. I expanded log4
by writing the radical using a rational
Ay
exponent and then applying the quotient rule, obtaining
1
log 4 x - log 4 y.
2
In Exercises 125–128, determine whether each statement is true
or false. If the statement is false, make the necessary change(s) to
produce a true statement.
125. ln22 =
126.
127.
128.
129.
130.
131.
ln 2
2
log7 49
= log7 49 - log7 7
log7 7
logb(x3 + y3) = 3 logb x + 3 logb y
5
logb(xy)5 = (logb x + logb y)
Use the change-of-base property to prove that
1
log e =
.
ln 10
If log 3 = A and log 7 = B, find log 7 9 in terms of A and B.
Write as a single term that does not contain a logarithm:
5
e ln 8x
- ln 2x2
.
132. If f(x) = log b x, show that
f(x + h) - f(x)
h h1
b , h ⬆ 0.
h
x
133. Use the proof of the product rule in the appendix to prove
the quotient rule.
= logb a1 +
Preview Exercises
Exercises 134–136 will help you prepare for the material covered
in the next section.
134. Solve for x: a(x - 2) = b(2x + 3).
135. Solve: x(x - 7) = 3.
1
x + 2
136. Solve:
= .
4x + 3
x
Section 3.4 Exponential and Logarithmic Equations
CHAPTER 3
Mid-Chapter Check Point
WHAT YOU KNOW: We evaluated and graphed
exponential functions [ f(x) = bx, b 7 0 and b ⬆ 1],
including the natural exponential function [f(x) = ex,
e ⬇ 2.718]. A function has an inverse that is a function
if there is no horizontal line that intersects the function’s
graph more than once. The exponential function passes
this horizontal line test and we called the inverse of the
exponential function with base b the logarithmic function
with base b. We learned that y = logb x is equivalent
to by = x. We evaluated and graphed logarithmic
functions, including the common logarithmic function
[f(x) = log 10 x or f(x) = log x] and the natural logarithmic
function [f(x) = log e x or f(x) = ln x]. We learned to use
transformations to graph exponential and logarithmic
functions. Finally, we used properties of logarithms to
expand and condense logarithmic expressions.
In Exercises 1–5, graph f and g in the same rectangular
coordinate system. Graph and give equations of all asymptotes.
Give each function’s domain and range.
1.
2.
3.
4.
5.
f(x)
f(x)
f(x)
f(x)
f(x)
=
=
=
=
=
2x and g(x) = 2x - 3
1 12 2 x and g(x) = 1 12 2 x - 1
ex and g(x) = ln x
log2 x and g(x) = log2(x - 1) + 1
log 1 x and g(x) = - 2 log 1 x
2
SECTION 3.4
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Objectives
Use like bases to solve
exponential equations.
Use logarithms to solve
exponential equations.
Use the definition of
a logarithm to solve
logarithmic equations.
Use the one-to-one
property of logarithms
to solve logarithmic
equations.
Solve applied problems
involving exponential and
logarithmic equations.
451
2
In Exercises 6–9, find the domain of each function.
6. f(x) = log3(x + 6)
8. h(x) = log3(x + 6)2
7. g(x) = log 3 x + 6
9. f(x) = 3x + 6
In Exercises 10–20, evaluate each expression without using a
calculator. If evaluation is not possible, state the reason.
10. log2 8 + log5 25
11. log3 19
12. log100 10
3
13. log 2 10
14. log2(log3 81) 15. log3 1 log2 18 2
log6 5
16. 6
17. ln e17
18. 10log 13
1p
19. log100 0.1
20. logpp
In Exercises 21–22, expand and evaluate numerical terms.
1xy
≤
22. ln(e19 x20)
21. log ¢
1000
In Exercises 23–25, write each expression as a single logarithm.
1
23. 8 log 7 x - log 7 y
24. 7 log 5 x + 2 log 5 x
3
1
25. ln x - 3 ln y - ln(z - 2)
2
26. Use the formulas
r nt
A = P a1 + b
and A = Pert
n
to solve this exercise. You decide to invest $8000 for 3 years
at an annual rate of 8%. How much more is the return if the
interest is compounded continuously than if it is compounded
monthly? Round to the nearest dollar.
Exponential and Logarithmic Equations
At age 20, you inherit $30,000. You’d like to put aside
$25,000 and eventually have over half a million
dollars for early retirement. Is this possible? In
Example 10 in this section, you will
see how techniques for solving
equations with variable
exponents provide an
answer to this question.
Exponential Equations
An exponential equation is an equation containing a variable in an exponent.
Examples of exponential equations include
23x - 8 = 16,
4x = 15, and 40e 0.6x = 240.
Some exponential equations can be solved by expressing each side of the equation
as a power of the same base. All exponential functions are one-to-one—that is, no
two different ordered pairs have the same second component. Thus, if b is a positive
number other than 1 and bM = bN, then M = N.