Coburn J.W., Herdlick J.D. College Algebra: Graphs and Models
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Properties of Logarithms
Given M, N, and are positive real numbers, and any real number p.
Product Property Quotient Property Power Property
The log of a product The log of a quotient is The log of a quantity
is a sum of logarithms. a difference of logarithms. to a power is the
power times the log
of the quantity.
CAUTION
䊳
It’s very important that you read and understand these properties correctly. In particular:
(1) , (2) , and (3) .
For and , statement (1) would indicate X, (2) would
indicate that X, and (3) would indicate that X.
In the statement of these properties, it’s worth reminding ourselves that the equal
sign “works both ways,” and we have . These properties
are often used to write a sum or difference of logarithmic terms as a single expression.
EXAMPLE 5
䊳
Rewriting Expressions Using Logarithmic Properties
Use the properties of logarithms to write each expression as a single term.
a. b. c.
Solution
䊳
a. product property
simplify
b. power property
product property
simplify
c. quotient property
Now try Exercises 27 through 42
䊳
We can verify that these properties produce equivalent results by entering the original
equation as Y
1
, the result after applying the properties as Y
2
, and viewing the results of
various inputs on a TABLE screen. Results for Example 5b (power and product prop-
erties) and 5c (quotient property) are shown in Figures 5.42 and 5.43 respectively.
log 1x ⫹ 22⫺ log x ⫽ log a
x ⫹ 2
x
b
⫽ ln3x
3
⫹ 6x
2
4
⫽ ln3x
2
1x ⫹ 624
2 ln x ⫹ ln
1x ⫹ 62⫽ ln x
2
⫹ ln 1x ⫹ 62
⫽ log
2
35
log
2
7 ⫹ log
2
5 ⫽ log
2
17
#
52
log 1x ⫹ 22⫺ log x2 ln x ⫹ ln
1x ⫹ 62log
2
7 ⫹ log
2
5
log
b
M ⫹ log
b
N ⫽ log
b
1MN2
2
1
⫽ 2 ⫺ 11 ⫽
2
1
log 110 ⫽ 2 ⫹ 1N ⫽ 10M ⫽ 100
log M
log N
⫽ log M ⫺ log Nlog a
M
N
b⫽
log M
log N
log1M ⫹ N2⫽ log M ⫹ log N
log
b
M
p
⫽ plog
b
Mlog
b
a
M
N
b⫽ log
b
M⫺log
b
Nlog
b
1MN2⫽ log
b
M⫹log
b
N
b ⫽ 1
520 CHAPTER 5 Exponential and Logarithmic Functions 5–42
College Algebra Graphs & Models—
WORTHY OF NOTE
For a more detailed verification of
these properties, see Appendix VI.
Figure 5.42
2 ln x ⫹ ln1x ⫹ 62⫽ ln1x
3
⫹ 6x
2
2
Figure 5.43
log 1x ⫹ 22⫺ log x ⫽ log 1
x ⫹ 2
x
2
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MM NNNNMM
EXAMPLE 6
䊳
Rewriting Logarithmic Expressions Using the Power Property
Use the power property of logarithms to rewrite each term as a product.
a. ln 5
x
b. c.
Solution
䊳
a. power property
b. power property
c. write radical using a rational exponent
power property
Now try Exercises 43 through 50
䊳
For examples of how these properties are used in context, see Exercises 73
and 74.
CAUTION
䊳
Note from Example 6b that parentheses must be used whenever the exponent is a sum
or difference. There is a huge difference between and .
Examples 5 and 6 illustrate how the properties of logarithms are used to consolidate
logarithmic terms, primarily in preparation for equation solving. In other cases, the
properties are used to rewrite or expand logarithmic expressions, so that certain other
procedures can be applied more easily. Example 7 actually lays the foundation for
more advanced mathematical work.
EXAMPLE 7
䊳
Rewriting Expressions Using Logarithmic Properties
Use the properties of logarithms to write the following expressions as sums or
differences of simple logarithmic terms.
a. log(x
2
z) b.
Solution
䊳
a. product property
power property
b.
power property
quotient property
Now try Exercises 51 through 58
䊳
As you begin working with applications of logarithmic properties, it may help to have
them written on a separate note card. This will enable you to compare each step and
property as they are applied, remembering that “M” and “N” can represent any positive
number or any positive, real-valued expression (see Reinforcing Basic Concepts on
page 527). For Example 7(a) we then have
log1x
2
z2⫽ log x
2
⫹ log z
⫽
1
2
3ln x ⫺ ln1x ⫹ 524
⫽
1
2
lna
x
x ⫹ 5
b
write radical using
a rational exponent
ln
A
x
x ⫹ 5
⫽ ln a
x
x ⫹ 5
b
1
2
⫽ 2 log x ⫹ log z
log 1x
2
z2⫽ log x
2
⫹ log z
ln
A
x
x ⫹ 5
x ⫹ 2 log 321x ⫹ 22 log 32
⫽
1
2
log x
log1x
⫽ log x
1
2
log 32
x⫹2
⫽ 1x ⫹ 22 log 32
ln 5
x
⫽ x ln 5
log 1x
log 32
x⫹2
5–43 Section 5.4 Properties of Logarithms 521
College Algebra Graphs & Models—
B. You’ve just seen how
we can apply the product,
quotient, and power properties
of logarithms
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C. The Change-of-Base Formula
Although base-10 and base-e logarithms dominate the mathematical landscape, there are
many practical applications of other bases. Fortunately, a formula exists that will convert
any given base into either base 10 or base e. It’s called the change-of-base formula.
Change-of-Base Formula
For the positive real numbers M, a, and b, with ,
base 10 base e arbitrary base a
Proof of the Change-of-Base Formula
For , we have in exponential form. It follows that
take base-
a
logarithm of both sides
power property of logarithms
divide by log
a
b
substitute log
b
M
for
y
EXAMPLE 8
䊳
Using the Change-of-Base Formula to Evaluate Expressions
Find the value of each expression using the change-of-base formula. Answer in
exact form and approximate form using nine digits, then verify the result using the
original base. Note that either base 10 or base e can be used.
a. log
3
29 b. log
5
3.6
Solution
䊳
a. b.
Check: Check:
Now try Exercises 59 through 66
䊳
The change-of-base formula can also be used to study and graph logarithmic func-
tions of any base. For , the right-hand expression is simply rewritten using
the formula and the equivalent function is . The new function can then be
evaluated as in Example 8, or used to study the graph of for any base b. See
Exercises 67 through 70.
D. Solving Applications of Logarithms
We end this section with one additional application of logarithms. For all living things,
the concentration of hydrogen ions in a solution plays an important role as their pres-
ence or absence alters the environment of other molecules in the solution. This can
dramatically affect the functionality of the solution, or the ability of an organism to
survive. The concentration of hydrogen ions (in moles per liter) is commonly
expressed in terms of what is called the pH scale. A low pH number corresponds to
high hydrogen ion concentration, and the solution is said to be acidic. A high pH
corresponds to low hydrogen ion concentration, and the solution is said to be basic.
Since the range of values is so large, the pH scale is logarithmic, meaning each unit
change in the pH scale represents a 10-fold increase or decrease in the concentration of
y ⫽ log
b
x
y ⫽
log
x
log b
y ⫽ log
b
x
5
0.795888947
⫽ 3.6 ✓3
3.065044752
⫽ 29 ✓
0.795888947 3.065044752
log
5
3.6 ⫽
ln 3.6
ln 5
log
3
29 ⫽
log
29
log 3
log
b
M ⫽
log
a
M
log
a
b
y ⫽
log
a
M
log
a
b
y log
a
b ⫽ log
a
M
log
a
1b
y
2⫽ log
a
M
b
y
⫽ My ⫽ log
b
M
log
b
M ⫽
log
a
M
log
a
b
log
b
M ⫽
ln M
ln b
log
b
M ⫽
log
M
log b
a, b ⫽ 1
522 CHAPTER 5 Exponential and Logarithmic Functions 5–44
College Algebra Graphs & Models—
C. You’ve just seen how
we can apply the change-
of-base formula
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5–45 Section 5.4 Properties of Logarithms 523
College Algebra Graphs & Models—
hydrogen ions. For example, tomato juice, with a pH level of 2, is 10 times more acidic
than orange juice, with a pH level of 3, and 100 times more acidic than grape juice
(pH ⫽ 4; the lower the pH number, the higher the ion concentration). The pH values
range from 0 to 14, with pure water at being deemed “neutral” (neither basic
nor acidic). Measuring pH levels plays an important role in biology, chemistry, food
science, environmental science, medicine, oceanography, personal care products, and
many other areas. The number of hydrogen atoms is usually represented by the term
with the pH number defined as
EXAMPLE 9A
䊳
The Concentration of Hydrogen Atoms in Ocean Water
Ocean water has a pH number of near 7.9. What is the concentration of hydrogen
ions? Write the result in scientific notation.
Solution
䊳
Begin with the basic formula and work from there.
pH formula
substitute 7.9 for pH
multiply by
exponential form
result
The hydrogen ion concentration in ocean water is moles/liter.
EXAMPLE 9B
䊳
Finding the pH Level of an Apple
The concentration of hydrogen ions in an everyday apple is very near .
What is the pH level of an apple?
Solution
䊳
pH formula
substitute for H
⫹
result
An apple has a pH level near 3.1.
Now try Exercises 73 through 78
䊳
3.1
7.94 ⫻ 10
⫺4
⫽⫺log 37.94 ⫻ 10
⫺4
4
pH ⫽⫺log 3H
⫹
4
7.94 ⫻ 10
⫺4
1.26 ⫻ 10
⫺8
1.26 ⫻ 10
⫺8
⫽ H
⫹
10
⫺7.9
⫽ H
⫹
⫺1 ⫺7.9 ⫽ log 3H
⫹
4
7.9 ⫽⫺log 3H
⫹
4
pH ⫽⫺log 3H
⫹
4
pH ⫽⫺log 3H
⫹
4H
⫹
,
pH ⫽ 7
䊳
CONCEPTS AND VOCABULARY
Fill in each blank with the appropriate word or phrase. Carefully reread the section if needed.
5.4 EXERCISES
4. The statement is an example of
the ________ -of- ________ formula.
5. Use all factor pairs of 36 to illustrate the product
property of logarithms. For example, since
36 ⫽ 4 ⭈ 9, is log(4 ⭈ 9) ⫽ log 4 ⫹ log 9?
6. Use integer divisors of 24 to illustrate the
quotient property of logarithms. For
example, since , is
log ?12 ⫽ log
1
24
2
2⫽ log 24 ⫺ log 2
12 ⫽
24
2
log
e
10 ⫽
log
10
log e
1. For the solution process is most
efficient if we apply a base ______ logarithm to
both sides.
2. To solve we can combine
terms using the ________ property, or add
to both sides and use the ________
property.
3. Since logarithmic functions are not defined for all
real numbers, we should check all “solutions” for
________ roots.
ln1x ⫹ 32
ln 2x ⫺ ln1x ⫹ 32⫽ 0,
e
⫺0.02x⫹1
⫽ 10,
D. You’ve just seen how
we can solve applications of
logarithms
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524 CHAPTER 5 Exponential and Logarithmic Functions 5–46
College Algebra Graphs & Models—
䊳
DEVELOPING YOUR SKILLS
Solve each equation by applying fundamental
properties. Round to thousandths.
7. 8.
9. 10.
11. 12.
13. 14.
Solve each exponential equation. Write answers in exact
form and in approximate form rounded to four decimal
places.
15. 16.
17. 18.
19.
20.
Solve each logarithmic equation. Write answers in exact
form and in approximate form rounded to four decimal
places.
21. 22.
23.
24.
25.
26.
Use properties of logarithms to write each expression as
a single term.
27. 28.
29.
30.
31. 32.
33. 34.
35. 36.
37.
38.
39. 40.
41.
42. log
3
13x
2
⫹ 5x2⫺ log
3
x
log
5
1x
2
⫺ 2x2⫹ log
5
x
⫺1
log
9
2 ⫹ log
9
15log
2
7 ⫹ log
2
6
ln1x
2
⫺ 252⫺ ln1x ⫹ 52
ln1x
2
⫺ 42⫺ ln1x ⫹ 22
ln1x ⫹ 32⫺ ln1x ⫺ 12ln1x ⫺ 52⫺ ln x
log1x ⫺ 22⫺ log xlog x ⫺ log1x ⫹ 12
log
6
30 ⫺ log
6
10log
3
28 ⫺ log
3
7
log1x ⫺ 32⫹ log1x ⫹ 32
log1x ⫹ 12⫹ log1x ⫺ 12
ln1x ⫹ 22⫹ ln13x2ln12x2⫹ ln1x ⫺ 72
3
4
ln14x2⫺ 6.9 ⫽⫺5.1
1
2
ln12x ⫹ 52⫹ 3 ⫽ 3.2
⫺4 log12x2⫹ 9 ⫽ 3.6
⫺1.5 ⫽ 2 log15 ⫺ x2⫺ 4
⫺15 ⫽⫺8 ln13x2⫹ 73 ln1x ⫹ 42⫺ 5 ⫽ 3
250e
0.05x⫹1
⫹ 175 ⫽ 1175
⫺150 ⫽ 290.8 ⫺ 190e
⫺0.75x
10
2x
⫹ 27 ⫽ 19010
x⫹5
⫺ 228 ⫽⫺150
2 ⫺ 3e
0.4x
⫽⫺74e
x⫺2
⫹ 5 ⫽ 70
10
x
⫽ 0.02410
x
⫽ 18.197
e
x
⫽ 0.343e
x
⫽ 9.025
log x ⫽ 1.6log x ⫽
1
4
ln x ⫽
1
2
ln x ⫽ 3.4
Use the power property of logarithms to rewrite each
term as the product of some quantity times a
logarithmic term.
43. 44.
45. 46.
47. 48.
49. log
5
81 50. log
7
121
Use the properties of logarithms to write the following
expressions as sums or differences of simple logarithmic
terms.
51. log(a
3
b) 52. log(m
2
n)
53. 54.
55. 56.
57. 58.
Evaluate each expression using the change-of-base
formula and either base 10 or base e. Answer in exact
form and in approximate form using nine decimal
places, then verify the result using the original base.
59. log
7
60 60. log
8
92
61. log
5
152 62. log
6
200
63. log
3
1.73205 64. log
2
1.41421
65. log
0.5
0.125 66. log
0.2
0.008
Use the change-of-base formula to write an equivalent
function, then evaluate the function as indicated (round
to six decimal places). Investigate and discuss any
patterns you notice in the output values, then determine
the next input that will continue the pattern.
67.
68.
69.
70. H1x2⫽ log
x; H1122, H122, H122
3
2
h1x2⫽ log
9
x; h122, h142, h182
g1x2⫽ log
2
x; g152, g1102, g1202
f
1x2⫽ log
3
x; f 152, f 1152, f 1452
log
a
B
3
3 ⫺ v
2v
blog a
A
x ⫺ 2
x
b
ln
a
m
2
n
3
bln a
x
2
y
b
ln
11
3
pq2ln 1x1
4
y2
log 2
3
34log122
ln 10
3x⫹2
ln 5
2x⫺1
log 15
x⫺3
log 8
x⫹2
䊳
WORKING WITH FORMULAS
71.
Use the change-of-base formula to verify the
“formula” shown.
log
b
M ⴝ
1
log
M
b
72. log
B
A log
C
B log
D
C ⴝ log
D
A
Use the change-of-base formula to verify the
“formula” shown.
##
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5–47 Section 5.4 Properties of Logarithms 525
College Algebra Graphs & Models—
䊳
APPLICATIONS
73. Pareto’s 80/20 principle: After observing that
80% of the land in his native Italy was owned by
20% of the population, Italian economist Vilfredo
Pareto (1848–1923) noted this disparity in many
other areas (20% of the workers produce 80% of
the output, 20% of the customers create 80% of the
revenue, etc.) and developed a mathematical model
for this phenomenon, called Pareto’s law. If N
represents the number of people with incomes
greater than X, then log log X,
where A and m are predetermined constants.
(a) Solve the equation for N and (b) given
and , find the number of people earning
over $200,000. Assume X is in hundreds of
thousands of dollars.
74. The species/area
relationship:
The study of
what is now
known as island
biogeography
originated with
Robert McArthur
and Edward O. Wilson in the 1960s. In general,
they found that the relationship between island area
and the number of species present could be
modeled by the equation log ,
where S represents the total number of species, A
represents the area of the island, while C and k are
predetermined constants that depend on the size
and proximity of other land masses as well as other
factors. This makes it possible to predict the
number of species on an island, when little other
information is available. (a) Solve the equation for
S and (b) given and , find the
predicted number of species an island with area of
.
75. Blood plasma pH
levels: To be safe
and usable, the
blood plasma held
by blood banks must
have a pH level
between 7.35 and
7.45. Blood outside
of this normal range
can cause
disorientation,
behavioral changes, or even death. Using ion-
sensitive electrodes, a sample of blood plasma is
known to have a concentration of
. Is the plasma usable?
H
⫹
⫽ 4.786 ⫻ 10
⫺8
A ⫽ 2000 km
2
C ⫽ 8k ⫽ 0.81
S ⫽ log C ⫹ k log A
A ⫽ 9900
m ⫽ 1.5
N ⫽ log A ⫺ m
76. Fresh milk: As milk
begins to sour, there is
a corresponding
decrease in pH level.
Fresh milk has a pH
level of near 6.5. After
transport from farm to
market, a sample of
milk is tested using ion-
sensitive electrodes and
is found to have a
concentration of
.
Is this shipment of milk
still suitable for market?
77. Soil acidity:
Throughout many parts
of the Midwest, surface
soils are neutral to
slightly alkaline. While
a majority of crops
might prefer a pH
neutral soil ,
some crops thrive in
more acidic soils
(potatoes, strawberries,
others). For these
crops, elemental sulfur
is applied to help
decrease the pH level (the optimum pH level for
potato crops is near 5.2). Measurements of the soil
on a certain midwestern farm indicate a hydrogen
ion concentration of . Is the
soil ready for a potato crop to be planted?
78. Acidity of gastric juices: The normal pH value of
human gastric juice can vary from 1 to 3,
depending on genetics, diet, and other factors. The
acidity is designed to control
various harmful
microorganisms that a person
may ingest as they eat.
Drinking large quantities of
water before a meal can have
a dramatic effect on this pH
value, sometimes raising it
beyond the normal range to as high as 4 or 5,
making it possible for some harmful bacteria to
survive. If a hospital patient’s stomach fluid has a
hydrogen ion concentration of
, is the pH level within a
normal range?
H
⫹
⫽ 3.981 ⫻ 10
⫺3
H
⫹
⫽ 1.259 ⫻ 10
⫺6
1pH ⫽ 72
H
⫹
⫽ 3.981 ⫻ 10
⫺5
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526 CHAPTER 5 Exponential and Logarithmic Functions 5–48
College Algebra Graphs & Models—
䊳
EXTENDING THE CONCEPT
79. Logarithmic properties can also be used to
compare the magnitude of very large numbers,
numbers too large for a handheld calculator to
manage. Use the power property of logarithms to
compare the numbers 600
601
and 601
600
. Which
number is larger?
80. Logarithmic properties can also be used to
compare the magnitude of very small numbers,
again numbers too small for a handheld calculator
to manage. Use a negative exponent and the power
property of logarithms to compare the numbers
and . Which number is smaller?
1
100
99
1
99
100
MID-CHAPTER CHECK
1. Write the following in logarithmic form.
a. b.
2. Write the following in exponential form.
a. b.
3. Solve each equation for the unknown:
a. b.
4. Solve each equation for the unknown:
a. b.
5. The homes in a popular neighborhood are growing
in value according to the formula
where t is the time in years, V
0
is the purchase
price of the home, and V(t) is the current value of
the home. (a) In 3 yr, how much will a $50,000
home be worth? (b) Use the TABLE feature of
your calculator to estimate how many years (to the
nearest year) until the home doubles in value.
6. The graph of the function has been
shifted right 3 units, up 2 units, and stretched by a
factor of 4. What is the equation of the resulting
function?
f
1x2⫽ 5
x
V1t2⫽ V
0
1
9
8
2
t
,
log
b
125 ⫽ 3log
27
x ⫽
1
3
1
1
3
2
4b
⫽ 9
2b⫺5
4
2x
⫽ 32
x⫺1
log
1296
6 ⫽ 0.25log
8
32 ⫽
5
3
81
5
4
⫽ 24327
2
3
⫽ 9
7. State the domain and range for
then find and state its domain and range.
Verify the inverse relationship using composition.
8. Write the following equations in logarithmic form,
then verify the result on a calculator.
a. b.
9. Write the following equations in exponential form,
then verify the result on a calculator.
a. b.
10. On August 15, 2007, an earthquake measuring 8.0
on the Richter scale struck coastal Peru. On
October17, 1989, right before Game 3 of the World
Series between the Oakland A’s and the San
Francisco Giants, the Loma Prieta earthquake,
measuring 7.1 on the Richter scale, struck the San
Francisco Bay area. How much more intense was
the Peruvian earthquake?
1.4 ln 4.0552
2
3
⫽ log
27
9
e
4
54.59881 ⫽ 3
4
f
⫺1
1x2
f
1x2⫽ 1x ⫺ 3 ⫹ 1,
䊳
MAINTAINING YOUR SKILLS
81. (4.4/4.5) State the zeroes of f and the equation of
any horizontal or vertical asymptotes given
.
82. (1.4) A sports shop can stock 36 cans of tennis
balls on shelves that are 9 in. deep, and 54 cans on
12-in. shelves. Assuming the relationship is linear,
(a) find the equation relating shelf size to number
of cans, and (b) use it to determine what size shelf
should be used to stock a full shipment of 72 cans.
f
1x2⫽
x
2
⫺ x ⫺ 6
x
2
⫺ 1
83. (R.4) Find all values of x that make the equation
true: .
84. (2.2) Determine the
equation of the function
shown in (a) shifted form
and (b) standard form
2x1x ⫺ 32⫹ 4 ⫽ 13x ⫹ 521x ⫺ 12
x
y
56784321
⫺2⫺1
⫺1
⫺2
⫺3
⫺4
⫺5
2
3
4
5
1
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5–49 Section 5.5 Solving Exponential and Logarithmic Equations 527
College Algebra Graphs & Models—
22 33 55 66 55 11
22 33 44 55 6611
Understanding Properties of Logarithms
To effectively use the properties of logarithms as a mathematical tool, a student must attain some degree of comfort and
fluency in their application. Otherwise we are resigned to using them as a template or formula, leaving little room for
growth or insight. This feature is designed to promote an understanding of the product and quotient properties of loga-
rithms, which play a role in the solution of logarithmic and exponential equations.
We begin by looking at some logarithmic expressions that are obviously true:
Next, we view the same expressions with their value understood mentally, illustrated by the numbers in the
background, rather than expressly written.
This will make the product and quotient properties of equality much easier to “see.” Recall the product property states:
and the quotient property states: Consider the following.
which is the same as saying which is the same as saying
Exercise 1: Repeat this exercise using logarithms of base 3 and various sums and differences.
Exercise 2: Use the basic concept behind these exercises to combine these expressions: (a)
(b) and (c) log1x2⫺ log1x ⫹ 32.ln1x ⫹ 22⫹ ln1x ⫺ 22,
log1x2⫹ log1x ⫹ 32,
log
b
M ⫺ log
b
N ⫽ log
b
a
M
N
blog
b
M ⫹ log
b
N ⫽ log
b
1MN2
1since
64
32
⫽ 221since 4
#
8 ⫽ 322
log
2
64⫺ log
2
32⫽ log
2
1
64
32
2log
2
4⫹ log
2
8⫽ log
2
14
#
82
log
2
64 ⫺ log
2
32 ⫽ log
2
2log
2
4 ⫹ log
2
8 ⫽ log
2
32
log
b
M ⫺ log
b
N ⫽ log
b
a
M
N
b.log
b
M ⫹ log
b
N ⫽ log
b
1MN2
log
2
64log
2
32log
2
16log
2
8log
2
4log
2
2
log
2
64 ⫽ 6log
2
32 ⫽ 5log
2
16 ⫽ 4
log
2
8 ⫽ 3log
2
4 ⫽ 2log
2
2 ⫽ 1
REINFORCING BASIC CONCEPTS
5.5 Solving Exponential and Logarithmic Equations
In this section, we’ll develop the ability to solve more general logarithmic and expo-
nential equations. A logarithmic equation has at least one term that involves the loga-
rithm of a variable. Likewise, an exponential equation is one that involves a variable
exponent on some base. In the same way that we might square both sides or divide both
sides of an equation in the solution process, we’ll show that we can also exponentiate
both sides or take logarithms of both sides to help obtain a solution.
A. Solving Logarithmic and Exponential Equations
One of the most common mistakes in solving exponential and logarithmic equations is
to apply the inverse function too early—before the equation has been simplified. Just
as we would naturally try to combine like terms for the equation
(prior to squaring both sides), the logarithmic terms in log must
be combined prior to applying the exponential form. In addition, since the domain of
is , logarithmic equations can sometimes produce extraneous roots,
and checking all answers is a good practice. We’ll illustrate by solving the equation
.log x ⫹ log
1x ⫹ 32⫽ 1
x 7 0y ⫽ log
b
x
x ⫹ log1x ⫹ 32⫽ 1
21x
⫹ 71x ⫽ 69
LEARNING OBJECTIVES
In Section 5.5 you will see
how we can:
A. Solve general logarithmic
and exponential
equations
B. Solve applications
involving logistic,
exponential, and
logarithmic functions
cob19545_ch05_517-527.qxd 9/2/10 9:43 PM Page 527
College Algebra Graphs & Models—
528 CHAPTER 5 Exponential and Logarithmic Functions 5–50
⫺3
3
0
5
EXAMPLE 1
䊳
Solving a Logarithmic Equation
Solve for x and check your answer: .
䊲
Algebraic Solution
original equation
product property
exponential form,
distribute
x
set equal to 0
factor
result
x ⫽⫺5 or x ⫽ 2
1x ⫹ 521x ⫺ 22⫽ 0
x
2
⫹ 3x ⫺ 10 ⫽ 0
x
2
⫹ 3x ⫽ 10
1
log 3x 1x ⫹ 324⫽ 1
log x ⫹ log
1x ⫹ 32⫽ 1
log x ⫹ log
1x ⫹ 32⫽ 1
䊲
Graphical Solution
Using the intersection-of-
graphs method, we enter
and . From the domain
we know , indicating
the solution will occur in QI.
After graphing both functions
using the window shown, the
intersection method shows
the only solution is .x ⫽ 2
x 7 0
Y
2
⫽ 1
Y
1
⫽ log X ⫹ log1X ⫹ 32
Check: The “solution” is outside the domain and is ignored. For ,
original equation
substitute 2 for
x
simplify
product property
Property I
You could also use a calculator to verify directly.
Now try Exercises 7 through 14
䊳
If the simplified form of an equation yields a logarithmic term on both sides, the
uniqueness property of logarithms provides an efficient way to work toward a solu-
tion. Since logarithmic functions are one-to-one, we have
The Uniqueness Property of Logarithms
For positive real numbers m, n, and ,
1. 2. If ,
then
Equal bases imply equal arguments.
EXAMPLE 2
䊳
Solving Logarithmic Equations Using the Uniqueness Property
Solve each equation using the uniqueness property.
a. b.
䊲
Algebraic Solution
a.
properties of logarithms
uniqueness property
solve for
x
result
1
3
⫽ x
2 ⫽ 6x
x ⫹ 2 ⫽ 7x
log
1x ⫹ 22⫽ log 7x
log
1x ⫹ 22⫽ log 7 ⫹ log x
ln 87 ⫺ ln x ⫽ ln 29log
1x ⫹ 22⫽ log 7 ⫹ log x
log
b
m ⫽ log
b
nthen m ⫽ n
m ⫽ nIf log
b
m ⫽ log
b
n,
b ⫽ 1
log 5 ⫽ 1log 2 ⫹
log 10 ⫽ 1
log12
#
52⫽ 1
log 2 ⫹ log 5 ⫽ 1
log 2 ⫹ log12 ⫹ 32⫽ 1
log x ⫹ log1x ⫹ 32⫽ 1
x ⫽ 2x ⫽⫺5
䊲
Graphical Solution
Deciding which method to use (intersection or zeroes)
can depend on the simplicity or complexity of the
equation, how the equation is given, and/or which
method gives the clearest view of the point(s) of
intersection. Here, we opt to use the zeroes method.
cob19545_ch05_528-538.qxd 9/2/10 9:48 PM Page 528
a. After setting the
equation equal to
zero, we enter
and locate the
x-intercept (if it
exists) using
(CALC)
2:Zero. The graph
reveals an
x-intercept between 0 and 1, and using these values as
bounds we locate the zero at (Figure 5.44).
b. Once again
setting the
equation equal to
zero, we enter
29 to
locate the
x-intercept (if it
exists). The graph
shows an
x-intercept near 3, with the calculator indicating the
zero is exactly (Figure 5.45).x ⫽ 3
ln X ⫺ ln
Y
1
⫽ ln 87 ⫺
x ⫽
1
3
TRACE
2nd
log 7 ⫺ log X
Y
1
⫽ log1X ⫹ 22⫺
5–51 Section 5.5 Solving Exponential and Logarithmic Equations 529
College Algebra Graphs & Models—
⫺2
2
0
5
Figure 5.44
Now try Exercises 15 through 20
䊳
Often the solution may depend on using a variety of algebraic skills in addition to
logarithmic or exponential properties.
EXAMPLE 3
䊳
Solving Logarithmic Equations
Solve the equation and check your answer.
䊲
Algebraic Solution
given equation
quotient property
uniqueness property
clear denominator
set equal to 0
0 ⫽ x
2
⫹ 8x ⫺ 12
x ⫹ 12 ⫽ x
2
⫹ 9x
x ⫹ 12
x
⫽ x ⫹ 9
log
a
x ⫹ 12
x
b⫽ log
1x ⫹ 92
log
1x ⫹ 122⫺ log x ⫽ log 1x ⫹ 92
log
1x ⫹ 122⫺ log x ⫽ log 1x ⫹ 92
b.
quotient property
uniqueness property
clear denominator
result
3 ⫽ x
87 ⫽ 29x
87
x
⫽ 29
ln
a
87
x
b⫽ ln
29
ln 87 ⫺ ln
x ⫽ ln 29
⫺2
2
0
10
Figure 5.45
WORTHY OF NOTE
The uniqueness property can also
be viewed as exponentiating both
sides using the appropriate base,
then applying Property IV.
䊲
Graphical Solution
Using the intersection-of-graphs method, we
enter as Y
1
and
as Y
2
on the screen. Using
(CALC) 5:intersect, we find the
graphs intersect at , and that
this is the only solution (knowing the graphs’
basic shape, we conclude they cannot intersect
again).
x ⫽ 1.2915026
TRACE
2nd
Y=
log1X ⫹ 92
log
1X ⫹ 122⫺ log x
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