D. Solving Applications of Logarithms
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Section 5.4 Properties of Logarithms
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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 ϭ 4; the lower the pH number, the higher the ion concentration). The pH values
range from 0 to 14, with pure water at pH ϭ 7 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
Hϩ, with the pH number defined as pH ϭ Ϫlog 3H ϩ 4
EXAMPLE 9A
ᮣ
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
ᮣ
Begin with the basic formula and work from there.
pH ϭ Ϫlog 3 H ϩ 4
7.9 ϭ Ϫlog 3H ϩ 4
Ϫ7.9 ϭ log 3H ϩ 4
10Ϫ7.9 ϭ H ϩ
1.26 ϫ 10Ϫ8 ϭ H ϩ
pH formula
substitute 7.9 for pH
multiply by Ϫ1
exponential form
result
The hydrogen ion concentration in ocean water is 1.26 ϫ 10Ϫ8 moles/liter.
EXAMPLE 9B
ᮣ
Finding the pH Level of an Apple
The concentration of hydrogen ions in an everyday apple is very near 7.94 ϫ 10Ϫ4.
What is the pH level of an apple?
Solution
D. You’ve just seen how
we can solve applications of
logarithms
pH ϭ Ϫlog 3 H ϩ 4
ϭ Ϫlog 37.94 ϫ 10Ϫ4 4
Ϸ 3.1
ᮣ
pH formula
substitute 7.94 ϫ 10Ϫ4 for Hϩ
result
An apple has a pH level near 3.1.
Now try Exercises 73 through 78 ᮣ
5.4 EXERCISES
ᮣ
CONCEPTS AND VOCABULARY
Fill in each blank with the appropriate word or phrase. Carefully reread the section if needed.
1. For eϪ0.02xϩ1 ϭ 10, the solution process is most
efficient if we apply a base ______ logarithm to
both sides.
4. The statement loge 10 ϭ
2. To solve ln 2x Ϫ ln1x ϩ 32 ϭ 0, we can combine
terms using the ________ property, or add
ln1x ϩ 32 to both sides and use the ________
property.
5. Use all factor pairs of 36 to illustrate the product
property of logarithms. For example, since
36 ϭ 4 и 9, is log(4 и 9) ϭ log 4 ϩ log 9?
3. Since logarithmic functions are not defined for all
real numbers, we should check all “solutions” for
________ roots.
log 10
is an example of
log e
the ________ -of- ________ formula.
6. Use integer divisors of 24 to illustrate the
quotient property of logarithms. For
example, since 12 ϭ 24
2 , is
2
ϭ
log
24 Ϫ log 2?
log 12 ϭ log 1 24
2
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CHAPTER 5 Exponential and Logarithmic Functions
DEVELOPING YOUR SKILLS
Solve each equation by applying fundamental
properties. Round to thousandths.
7. ln x ϭ 3.4
8. ln x ϭ 12
9. log x ϭ 14
10. log x ϭ 1.6
11. ex ϭ 9.025
12. ex ϭ 0.343
13. 10x ϭ 18.197
14. 10x ϭ 0.024
Solve each exponential equation. Write answers in exact
form and in approximate form rounded to four decimal
places.
15. 4exϪ2 ϩ 5 ϭ 70
16. 2 Ϫ 3e0.4x ϭ Ϫ7
17. 10xϩ5 Ϫ 228 ϭ Ϫ150 18. 102x ϩ 27 ϭ 190
19. Ϫ150 ϭ 290.8 Ϫ 190eϪ0.75x
20. 250e0.05xϩ1 ϩ 175 ϭ 1175
Solve each logarithmic equation. Write answers in exact
form and in approximate form rounded to four decimal
places.
21. 3 ln1x ϩ 42 Ϫ 5 ϭ 3
24. Ϫ4 log12x2 ϩ 9 ϭ 3.6
1
2
ln12x ϩ 52 ϩ 3 ϭ 3.2
26.
3
4
ln14x2 Ϫ 6.9 ϭ Ϫ5.1
Use properties of logarithms to write each expression as
a single term.
27. ln12x2 ϩ ln1x Ϫ 72
28. ln1x ϩ 22 ϩ ln13x2
29. log1x ϩ 12 ϩ log1x Ϫ 12
30. log1x Ϫ 32 ϩ log1x ϩ 32
31. log328 Ϫ log37
32. log630 Ϫ log610
33. log x Ϫ log1x ϩ 12
34. log1x Ϫ 22 Ϫ log x
35. ln1x Ϫ 52 Ϫ ln x
36. ln1x ϩ 32 Ϫ ln1x Ϫ 12
44. log 15xϪ3
45. ln 52xϪ1
46. ln 103xϩ2
47. log 122
3
48. log 2
34
49. log581
50. log7121
Use the properties of logarithms to write the following
expressions as sums or differences of simple logarithmic
terms.
51. log(a3b)
52. log(m2n)
4
53. ln 1x 1
y2
55. ln a
3
54. ln 1 1
pq2
x2
b
y
57. log a
56. ln a
xϪ2
b
A x
m2
b
n3
58. log a
3Ϫv
b
B 2v
3
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. log760
60. log892
61. log5152
62. log6200
63. log31.73205
64. log21.41421
65. log0.50.125
66. log0.20.008
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. f 1x2 ϭ log3 x; f 152, f 1152, f 1452
68. g1x2 ϭ log2 x; g152, g1102, g1202
37. ln1x2 Ϫ 42 Ϫ ln1x ϩ 22
38. ln1x Ϫ 252 Ϫ ln1x ϩ 52
2
39. log27 ϩ log26
43. log 8xϩ2
22. Ϫ15 ϭ Ϫ8 ln13x2 ϩ 7
23. Ϫ1.5 ϭ 2 log15 Ϫ x2 Ϫ 4
25.
Use the power property of logarithms to rewrite each
term as the product of some quantity times a
logarithmic term.
40. log92 ϩ log915
41. log5 1x2 Ϫ 2x2 ϩ log5xϪ1
69. h1x2 ϭ log9 x; h122, h142, h182
70. H1x2 ϭ log x; H1 122, H122, H1 223 2
42. log3 13x2 ϩ 5x2 Ϫ log3x
ᮣ
WORKING WITH FORMULAS
71. logb M ؍
1
logM b
Use the change-of-base formula to verify the
“formula” shown.
72. log B A
# log B # log C ؍log
C
D
DA
Use the change-of-base formula to verify the
“formula” shown.
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Section 5.4 Properties of Logarithms
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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 N ϭ log A Ϫ m log X,
where A and m are predetermined constants.
(a) Solve the equation for N and (b) given m ϭ 1.5
and A ϭ 9900, find the number of people earning
over $200,000. Assume X is in hundreds of
thousands of dollars.
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 ionsensitive electrodes and
is found to have a
concentration of
Hϩ ϭ 3.981 ϫ 10Ϫ5.
Is this shipment of milk
still suitable for market?
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 S ϭ log C ϩ k log A,
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 k ϭ 0.81 and C ϭ 8, find the
predicted number of species an island with area of
A ϭ 2000 km2.
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 1pH ϭ 72 ,
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 H ϩ ϭ 1.259 ϫ 10Ϫ6. Is the
soil ready for a potato crop to be planted?
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 ionsensitive electrodes, a sample of blood plasma is
known to have a concentration of
H ϩ ϭ 4.786 ϫ 10Ϫ8. Is the plasma usable?
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
H ϩ ϭ 3.981 ϫ 10Ϫ3, is the pH level within a
normal range?
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ᮣ
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 600601 and 601600. Which
number is larger?
ᮣ
5–48
CHAPTER 5 Exponential and Logarithmic Functions
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
1
1
and
. Which number is smaller?
100
99
10099
MAINTAINING YOUR SKILLS
81. (4.4/4.5) State the zeroes of f and the equation of
any horizontal or vertical asymptotes given
x2 Ϫ x Ϫ 6
.
f 1x2 ϭ
x2 Ϫ 1
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.
83. (R.4) Find all values of x that make the equation
true: 2x1x Ϫ 32 ϩ 4 ϭ 13x ϩ 52 1x Ϫ 12 .
84. (2.2) Determine the
equation of the function
shown in (a) shifted form
and (b) standard form
y
5
4
3
2
1
Ϫ2Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
1 2 3 4 5 6 7 8 x
MID-CHAPTER CHECK
1. Write the following in logarithmic form.
2
5
a. 273 ϭ 9
b. 814 ϭ 243
2. Write the following in exponential form.
a. log832 ϭ 53
b. log12966 ϭ 0.25
3. Solve each equation for the unknown:
a. 42x ϭ 32xϪ1
b. 1 13 2 4b ϭ 92bϪ5
4. Solve each equation for the unknown:
a. log27x ϭ 13
b. logb125 ϭ 3
5. The homes in a popular neighborhood are growing
in value according to the formula V1t2 ϭ V0 1 98 2 t,
where t is the time in years, V0 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 f 1x2 ϭ 5x has been
shifted right 3 units, up 2 units, and stretched by a
factor of 4. What is the equation of the resulting
function?
7. State the domain and range for f 1x2 ϭ 1x Ϫ 3 ϩ 1,
then find f Ϫ1 1x2 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. 81 ϭ 34
b. e4 Ϸ 54.598
9. Write the following equations in exponential form,
then verify the result on a calculator.
2
a. ϭ log279
b. 1.4 Ϸ ln 4.0552
3
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?
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Section 5.5 Solving Exponential and Logarithmic Equations
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REINFORCING BASIC CONCEPTS
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 logarithms, which play a role in the solution of logarithmic and exponential equations.
We begin by looking at some logarithmic expressions that are obviously true:
log22 ϭ 1
log216 ϭ 4
log24 ϭ 2
log232 ϭ 5
log28 ϭ 3
log264 ϭ 6
Next, we view the same expressions with their value understood mentally, illustrated by the numbers in the
background, rather than expressly written.
1
2
log22
3
log24
log28
4
log216
5
log232
6
log264
This will make the product and quotient properties of equality much easier to “see.” Recall the product property states:
M
logb M ϩ logb N ϭ logb 1MN2 and the quotient property states: logb M Ϫ logb N ϭ logb a b. Consider the following.
N
log24 ϩ log28 ϭ log232
2
3
5
which is the same as saying
log24 ϩ log28 ϭ log2 14 # 82
1since 4 # 8 ϭ 322
logb M ϩ logb N ϭ logb 1MN2
log264 Ϫ log232 ϭ log22
6
5
1
which is the same as saying
log264 Ϫ log232 ϭ log2 1 64
32 2
1since 64
32 ϭ 22
M
logb M Ϫ logb N ϭ logb a b
N
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) log1x2 ϩ log1x ϩ 32,
(b) ln1x ϩ 22 ϩ ln1x Ϫ 22, and (c) log1x2 Ϫ log1x ϩ 32.
5.5
Solving Exponential and Logarithmic Equations
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
In this section, we’ll develop the ability to solve more general logarithmic and exponential equations. A logarithmic equation has at least one term that involves the logarithm 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 21x ϩ 7 1x ϭ 69
(prior to squaring both sides), the logarithmic terms in log x ϩ log1x ϩ 32 ϭ 1 must
be combined prior to applying the exponential form. In addition, since the domain of
y ϭ logb x is x 7 0, logarithmic equations can sometimes produce extraneous roots,
and checking all answers is a good practice. We’ll illustrate by solving the equation
log x ϩ log 1x ϩ 32 ϭ 1.
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EXAMPLE 1
ᮢ
5–50
CHAPTER 5 Exponential and Logarithmic Functions
ᮣ
Solving a Logarithmic Equation
Solve for x and check your answer: log x ϩ log 1x ϩ 32 ϭ 1.
ᮢ
Algebraic Solution
log x ϩ log 1x ϩ 32 ϭ 1
log 3x 1x ϩ 32 4 ϭ 1
x2 ϩ 3x ϭ 101
x2 ϩ 3x Ϫ 10 ϭ 0
1x ϩ 52 1x Ϫ 22 ϭ 0
x ϭ Ϫ5 or x ϭ 2
original equation
product property
exponential form,
distribute x
set equal to 0
factor
result
Graphical Solution
Using the intersection-ofgraphs method, we enter
Y1 ϭ log X ϩ log1X ϩ 32
and Y2 ϭ 1. From the domain
we know x 7 0, indicating
the solution will occur in QI.
After graphing both functions
using the window shown, the
intersection method shows
the only solution is x ϭ 2.
3
0
5
Ϫ3
Check: The “solution” x ϭ Ϫ5 is outside the domain and is ignored. For x ϭ 2,
log x ϩ log1x ϩ 32 ϭ 1 original equation
log 2 ϩ log12 ϩ 32 ϭ 1 substitute 2 for x
log 2 ϩ log 5 ϭ 1 simplify
log12 # 52 ϭ 1 product property
log 10 ϭ 1 Property I
You could also use a calculator to verify log 2 ϩ log 5 ϭ 1 directly.
Now try Exercises 7 through 14
ᮣ
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 solution. Since logarithmic functions are one-to-one, we have
The Uniqueness Property of Logarithms
For positive real numbers m, n, and b 1,
1. If logb m ϭ logb n,
2. If
m
then logbm
then m ϭ n
Equal bases imply equal arguments.
EXAMPLE 2
ᮣ
n,
logbn
Solving Logarithmic Equations Using the Uniqueness Property
Solve each equation using the uniqueness property.
a. log 1x ϩ 22 ϭ log 7 ϩ log x
b. ln 87 Ϫ ln x ϭ ln 29
ᮢ
ᮢ
Algebraic Solution
a. log 1x ϩ 22
log 1x ϩ 22
xϩ2
2
1
3
ϭ log 7 ϩ log x
ϭ log 7x
ϭ 7x
ϭ 6x
ϭx
properties of logarithms
uniqueness property
solve for x
result
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.