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D. Solving Applications of Logarithms

D. Solving Applications of Logarithms

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12:35 AM

Page 523

College Algebra Graphs & Models—


Section 5.4 Properties of Logarithms


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


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.


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


The hydrogen ion concentration in ocean water is 1.26 ϫ 10Ϫ8 moles/liter.


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?


D. You’ve just seen how

we can solve applications of


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


An apple has a pH level near 3.1.

Now try Exercises 73 through 78 ᮣ



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 ________


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




24 Ϫ log 2?

log 12 ϭ log 1 24




4:54 PM

Page 524

College Algebra Graphs & Models—



CHAPTER 5 Exponential and Logarithmic Functions


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


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


21. 3 ln1x ϩ 42 Ϫ 5 ϭ 3

24. Ϫ4 log12x2 ϩ 9 ϭ 3.6



ln12x ϩ 52 ϩ 3 ϭ 3.2




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


48. log 2


49. log581

50. log7121

Use the properties of logarithms to write the following

expressions as sums or differences of simple logarithmic


51. log(a3b)

52. log(m2n)


53. ln 1x 1


55. ln a


54. ln 1 1





57. log a

56. ln a



A x




58. log a



B 2v


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


39. log27 ϩ log26

43. log 8xϩ2

22. Ϫ15 ϭ Ϫ8 ln13x2 ϩ 7

23. Ϫ1.5 ϭ 2 log15 Ϫ x2 Ϫ 4


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


71. logb M ‫؍‬


logM b

Use the change-of-base formula to verify the

“formula” shown.

72. log B A

# log B # log C ‫ ؍‬log




Use the change-of-base formula to verify the

“formula” shown.



12:36 AM

Page 525

College Algebra Graphs & Models—


Section 5.4 Properties of Logarithms



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


The study of

what is now

known as island


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


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?



9:43 PM

Page 526

College Algebra Graphs & Models—



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?


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




. Which number is smaller?





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













1 2 3 4 5 6 7 8 x


1. Write the following in logarithmic form.



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


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.


a. ϭ log279

b. 1.4 Ϸ ln 4.0552


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?



9:43 PM

Page 527

College Algebra Graphs & Models—


Section 5.5 Solving Exponential and Logarithmic Equations



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.













This will make the product and quotient properties of equality much easier to “see.” Recall the product property states:


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.


log24 ϩ log28 ϭ log232




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




which is the same as saying

log264 Ϫ log232 ϭ log2 1 64

32 2

1since 64

32 ϭ 22


logb M Ϫ logb N ϭ logb a b


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.


Solving Exponential and Logarithmic Equations


In Section 5.5 you will see

how we can:

A. Solve general logarithmic

and exponential


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.



9:48 PM

Page 528

College Algebra Graphs & Models—




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



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.





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


then logbm

then m ϭ n

Equal bases imply equal arguments.




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





ϭ log 7 ϩ log x

ϭ log 7x

ϭ 7x

ϭ 6x


properties of logarithms

uniqueness property

solve for x


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.

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