B. Applications of Logistic, Exponential, and Logarithmic Functions
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CHAPTER 5 Exponential and Logarithmic Functions
The company originally had only a 6% market share.
M1302 ϭ
66
substitute 30 for t
1 ϩ 10eϪ0.051302
66
ϭ
1 ϩ 10eϪ1.5
Ϸ 20.4
simplify
result
After 30 days, they held a 20.4% market share.
b. For part (b), we replace M(t) with 60 and
solve for t.
66
1 ϩ 10eϪ0.05t
6011 ϩ 10eϪ0.05t 2 ϭ 66
1 ϩ 10eϪ0.05t ϭ 1.1
10eϪ0.05t ϭ 0.1
eϪ0.05t ϭ 0.01
ln eϪ0.05t ϭ ln 0.01
Ϫ0.05t ϭ ln 0.01
ln 0.01
tϭ
Ϫ0.05
Ϸ 92
60 ϭ
given
multiply by 1 ϩ 10eϪ0.05t
divide by 60
subtract 1
divide by 10
b. Using the intersection-of-graphs method, we
graph Y1 with Y2 ϭ 60 to find any point(s) of
intersection. For the window size, we reason
that after 30 days, there is only a 20.4% market
share (x must be much greater than 30), and a
60% market share is being explored (y must be
greater than 60). Using the window indicated in
Figure 5.51 reveals that a 60% market share
will be attained shortly after the 92nd day.
apply base-e logarithms
Figure 5.51
Property III
80
solve for t (exact form)
approximate form
0
120
0
The company will reach a 60% market share in about 92 days.
Now try Exercises 49 and 50
ᮣ
P0
b to find an
P
altitude H, given a temperature and the atmospheric (barometric) pressure in centimeters of mercury (cmHg). Using the tools from this section, we are now able to find the
atmospheric pressure for a given altitude and temperature.
Earlier we used the barometric equation H ϭ 130T ϩ 80002 ln a
EXAMPLE 8
ᮣ
Using Logarithms to Determine Atmospheric Pressure
Suppose a group of climbers has just scaled Mt. Rainier, the highest mountain of
the Cascade Range in western Washington State. If the mountain is about 4395 m
high and the temperature at the summit is Ϫ22.5°C, what is the atmospheric
pressure at this altitude? The pressure at sea level is P0 ϭ 76 cmHg.
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Section 5.5 Solving Exponential and Logarithmic Equations
ᮢ
Algebraic Solution
H ϭ 130T ϩ 80002 ln a
P0
b
P
given
4395 ϭ 3 301Ϫ22.52 ϩ 8000 4 ln a
4395 ϭ 7325 ln a
0.6 ϭ ln a
76
b
P
76
b
P
substitute 4395
for H, 76 for P0,
and Ϫ22.5 for T
simplify
76
b
P
divide by 7325
76
P
0.6
Pe ϭ 76
76
P ϭ 0.6
e
Ϸ 41.7
e0.6 ϭ
exponential form
multiply by P
divide by e 0.6
(exact form)
approximate form
B. You’ve just seen how
we can solve applications
involving logistic, exponential,
and logarithmic functions
535
Graphical Solution
To ensure that no algebraic errors are introduced, we’ll
enter the function as it appears after the substitutions
are made:
76
Y1 ϭ 3301Ϫ22.52 ϩ 80004 ln a b.
x
For the window size,
6000
we reason that since
x must be between 0
and 76, and y is
equal to 4395, we
0
100
only need the first
Quadrant and a
“frame” around the
window that allows a
Ϫ1000
clear view of the
intersection point. Using the window indicated shows that
at an altitude of 4395 m and a temperature of Ϫ22.5°C,
the atmospheric pressure is about 41.7 cmHg.
Now try Exercises 53 and 54 ᮣ
Additional applications involving appreciation/depreciation, Newton’s law of cooling,
space ship velocities and more, can be found in the Exercise set. See Exercises 55
through 66.
5.5 EXERCISES
ᮣ
CONCEPTS AND VOCABULARY
Fill in each blank with the appropriate word or phrase. Carefully reread the section if needed.
1. The expression log2x represents a
term,
while the expression log29 represents a
term.
2. To solve the equation ln1x ϩ 32 Ϫ ln x ϭ 7, we
like terms using logarithmic properties,
prior to writing the equation in
form.
3. If certain conditions are met, we know if
logb M ϭ logb N, then M ϭ N. This is a statement
of the
property, which is valid since
logarithmic functions are
-to.
4. Since the domain of y ϭ logb x is
. solving
logarithmic equations will sometimes produce
roots. Checking all solutions to
logarithmic equations is a necessary step.
5. Answer true or false and explain your response:
6. Answer true or false and explain your response:
logb 1M ϩ N2 ϭ logb 1M2 ϩ logb 1N2
logb a
logb M
M
bϭ
N
logb N
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CHAPTER 5 Exponential and Logarithmic Functions
DEVELOPING YOUR SKILLS
Solve each equation and check your answers.
7. log 4 ϩ log1x Ϫ 72 ϭ 2
26. ln 5 ϩ ln1x Ϫ 22 ϭ 1
27. log1x ϩ 82 ϩ log x ϭ log1x ϩ 182
8. log 5 ϩ log1x Ϫ 92 ϭ 1
9. log12x Ϫ 52 Ϫ log 78 ϭ Ϫ1
10. log14 Ϫ 3x2 Ϫ log 145 ϭ Ϫ2
11. log1x Ϫ 152 Ϫ 2 ϭ Ϫlog x
12. log x Ϫ 1 ϭ Ϫlog 1x Ϫ 92
13. log 12x ϩ 12 ϭ 1 Ϫ log x
14. log 13x Ϫ 132 ϭ 2 Ϫ log x
Solve each equation using the uniqueness property.
15. log 15x ϩ 22 ϭ log 2
16. log 12x Ϫ 32 ϭ log 3
17. log4 1x ϩ 22 Ϫ log43 ϭ log4 1x Ϫ 12
18. log3 1x ϩ 62 Ϫ log3 x ϭ log3 5
19. ln 18x Ϫ 42 ϭ ln 2 ϩ ln x
20. ln 1x Ϫ 12 ϩ ln 6 ϭ ln 13x2
Solve each equation using any appropriate method.
State solutions in both exact form and in approximate
form rounded to four decimal places. Clearly identify
any extraneous roots. If there are no solutions, so state.
28. log1x ϩ 142 Ϫ log x ϭ log1x ϩ 62
29. ln12x ϩ 12 ϭ 3 ϩ ln 6
30. ln 21 ϭ 1 ϩ ln1x Ϫ 22
31. log1Ϫx Ϫ 12 ϭ log15x2 ϩ log x
32. log11 Ϫ x2 ϩ log x ϭ log1x ϩ 42
33. log1x Ϫ 12 Ϫ log x ϭ log1x Ϫ 32
34. ln x ϩ ln1x Ϫ 22 ϭ ln 4
35. 7x ϭ 231
36. 6x ϭ 3589
37. 53x Ϫ 2 ϭ 128,965
38. 93x Ϫ 3 ϭ 78,462
39. 2xϩ1 ϭ 3x
40. 7x ϭ 42xϪ1
Solve each equation using the zeroes method or the
intersection-of-graphs method. Round approximate
solutions to three decimal places.
3
41. 2
x ϭ ln1x ϩ 52
42.
x2 Ϫ 25
ϭ Ϫln1x ϩ 92 ϩ 6
x2 Ϫ 9
43. 2x
2
ϪxϪ6
ϭ x2 ϩ x Ϫ 6
21. log12x Ϫ 12 ϩ log 5 ϭ 1
1
44. x3 Ϫ 9x ϭ ex
2
22. log1x Ϫ 72 ϩ log 3 ϭ 2
45.
250
ϭ 200
1 ϩ 4eϪ0.06x
24. log3 1x Ϫ 42 ϩ log3 172 ϭ 2
46.
80
ϭ 50
1 ϩ 15eϪ0.06x
23. log2 192 ϩ log2 1x ϩ 32 ϭ 3
25. ln1x ϩ 72 ϩ ln 9 ϭ 2
ᮣ
WORKING WITH FORMULAS
47. Logistic growth: P1t2 ؍
C
1 ؉ ae؊kt
For populations that exhibit logistic growth, the
population at time t is modeled by the function
shown, where C is the carrying capacity of the
population (the maximum population that can be
supported over a long period of time), k is the
P102
growth constant, and a ϭ C Ϫ
P102 . Solve the
formula for t, then use the result to find the value of
t given C ϭ 450, a ϭ 8, P ϭ 400, and k ϭ 0.075.
48. Estimating time of death: h ؍؊3.9 # lna
T ؊ TR
b
T0 ؊ TR
Using the formula shown, a forensic expert can
compute the approximate time of death for a
person found recently expired, where T is the body
temperature when it was found, TR is the (constant)
temperature of the room, T0 is the body
temperature at the time of death (T0 ϭ 98.6°F), and
h is the number of hours since death. If the body
was discovered at 9:00 A.M. with a temperature of
86.2°F, in a room at 73°F, at approximately what
time did the person expire? (Note this formula is a
version of Newton’s law of cooling.)
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Section 5.5 Solving Exponential and Logarithmic Equations
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APPLICATIONS
Use the barometric equation H ؍130T ؉ 80002 ln a
P0
b
P
for exercises 49 and 50. Recall that P0 ؍76 cmHg.
49. Altitude and temperature: A sophisticated spy
plane is cruising at an altitude of 18,250 m. If the
temperature at this altitude is Ϫ75°C, what is the
barometric pressure?
50. Altitude and temperature: A large weather
balloon is released and takes altitude, pressure, and
temperature readings as it climbs, and radios the
information back to Earth. What is the pressure
reading at an altitude of 5000 m, given the
temperature is Ϫ18°C?
51. Stocking a lake: A farmer wants to stock a private
lake on his property with catfish. A specialist studies
the area and depth of the lake, along with other
factors, and determines it can support a maximum
population of around 750 fish, with growth modeled
750
by the function P1t2 ϭ
, where P(t)
1 ϩ 24eϪ0.075t
gives the current population after t months. (a) How
many catfish did the farmer initially put in the
lake? (b) How many months until the population
reaches 300 fish?
52. Increasing sales: After expanding their area of
operations, a manufacturer of small storage
buildings believes the larger area can support sales
of 40 units per month. After increasing the
advertising budget and enlarging the sales force,
sales are expected to grow according to the model
40
S1t2 ϭ
, where S(t) is the expected
1 ϩ 1.5eϪ0.08t
number of sales after t months. (a) How many sales
were being made each month, prior to the expansion?
(b) How many months until sales reach 25 units per
month?
Use Newton’s law of cooling T ؍TR ؉ (T0 ؊ TR)ekh
to complete Exercises 57 and 58. Recall that water
freezes at 32؇F and use k ؍؊0.012. Refer to
Section 5.2, page 498 as needed.
53. Making popsicles: On a hot summer day, Sean
and his friends mix some Kool-Aid® and decide to
freeze it in an ice tray to make popsicles. If the
water used for the Kool-Aid® was 75°F and the
freezer has a temperature of Ϫ20°F, how long will
they have to wait to enjoy the treat?
54. Freezing time: Suppose the current temperature in
Esconabe, Michigan, was 47°F when a 5°F arctic
cold front moved over the state. How long would it
take a puddle of water to freeze over?
Depreciation/appreciation: As time passes, the value of
certain items decrease (appliances, automobiles, etc.),
while the value of other items increase (collectibles,
real estate, etc.). The time T in years for an item to
reach a future value can be modeled by the formula
Vn
T ؍k ln a b, where Vn is the purchase price when
Vf
new, Vf is its future value, and k is a constant that
depends on the item.
55. Automobile depreciation: If a new car is purchased
for $28,500, find its value 3 yr later if k ϭ 5.
56. Home appreciation: If a new home in an
“upscale” neighborhood is purchased for $130,000,
find its value 12 yr later if k ϭ Ϫ16.
Drug absorption: The time required for a certain
percentage of a drug to be absorbed by the body after
injection depends on the drug’s absorption rate. This
؊ln p
can be modeled by the function T( p) ؍
, where
k
p represents the percent of the drug that remains
unabsorbed (expressed as a decimal), k is the absorption
rate of the drug, and T( p) represents the elapsed time.
57. For a drug with an absorption rate of 7.2%, (a) find
the time required (to the nearest hour) for the body
to absorb 35% of the drug, and (b) find the percent
of this drug (to the nearest half percent) that
remains unabsorbed after 24 hr.
58. For a drug with an absorption rate of 5.7%, (a) find
the time required (to the nearest hour) for the body
to absorb 50% of the drug, and (b) find the percent
of this drug (to the nearest half percent) that
remains unabsorbed after 24 hr.
Spaceship velocity: In space travel, the change in the
velocity of a spaceship Vs (in km/sec) depends on the
mass of the ship Ms (in
tons), the mass of the fuel
which has been burned Mf
(in tons) and the escape
velocity of the exhaust Ve
(in km/sec). Disregarding
frictional forces, these are
related by the equation
Ms
b.
Vs ؍Ve ln a
Ms Ϫ Mf
59. For the Jupiter VII rocket, find the mass of the fuel
Mf that has been burned if Vs ϭ 6 km/sec when
Ve ϭ 8 km/sec, and the ship’s mass is 100 tons.
60. For the Neptune X satellite booster, find the mass
of the ship Ms if Mf ϭ 75 tons of fuel has been
burned when Vs ϭ 8 km/sec and Ve ϭ 10 km/sec.
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Learning curve: The job performance of a new
employee when learning a repetitive task (as on an
assembly line) improves very quickly at first, then
grows more slowly over time. This can be modeled by
the function P(t) ؍a ؉ b ln t, where a and b are
constants that depend on the type of task and the
training of the employee.
61. The number of toy planes an employee can
assemble from its component parts depends on the
length of time the employee has been working.
This output is modeled by P1t2 ϭ 5.9 ϩ 12.6 ln t,
where P(t) is the number of planes assembled daily
after working t days. (a) How many planes is an
ᮣ
employee making after 5 days on the job? (b) How
many days until the employee is able to assemble
34 planes per day?
62. The number of circuit boards an associate can
assemble from its component parts depends on the
length of time the associate has been working. This
output is modeled by B1t2 ϭ 1 ϩ 2.3 ln t, where
B(t) is the number of boards assembled daily after
working t days. (a) How many boards is an
employee completing after 9 days on the job?
(b) How long will it take until the employee is able
to complete 10 boards per day?
EXTENDING THE CONCEPT
Solve the following equations. Note that equations Exercises 63 and 64 are in quadratic form.
63. 2e2x Ϫ 7ex ϭ 15
64. 3e2x Ϫ 4ex Ϫ 7 ϭ Ϫ3
65. Use the algebraic method to find the inverse
function.
a. f 1x2 ϭ 2xϩ1
b. y ϭ 2 ln 1x Ϫ 32
66. Show that g1x2 ϭ f Ϫ1 1x2 by composing the
functions.
a. f 1x2 ϭ 3xϪ2; g1x2 ϭ log3 x ϩ 2
b. f 1x2 ϭ exϪ1; g1x2 ϭ ln x ϩ 1
67. Use properties of logarithms and/or exponents to
show
a. y ϭ 2x is equivalent to y ϭ ex ln 2.
b. y ϭ bx is equivalent to y ϭ erx,
where r ϭ ln b.
68. Use test values for p and q to demonstrate that the
following relationships are false.
a. ln 1pq2 ϭ ln p ln q b. ln p ϩ ln q ϭ ln1p ϩ q2
p
ln p
c. ln a b ϭ
q
ln q
69. Match each equation with the most appropriate solution strategy, and justify/discuss why.
a. exϩ1 ϭ 25
apply base-10 logarithm to both sides
b. log12x ϩ 32 ϭ log 53
rewrite and apply uniqueness property for exponentials
c. log1x2 Ϫ 3x2 ϭ 2
apply uniqueness property for logarithms
2x
d. 10 ϭ 97
apply either base-10 or base-e logarithm
5xϪ3
ϭ 32
e. 2
apply base-e logarithm
xϩ2
ϭ 23
f. 7
write in exponential form
ᮣ
MAINTAINING YOUR SKILLS
70. (3.3) Match the graph shown
with its correct equation,
without actually graphing
the function.
a. y ϭ x2 ϩ 4x Ϫ 5
b. y ϭ Ϫx2 Ϫ 4x ϩ 5
c. y ϭ Ϫx2 ϩ 4x ϩ 5
d. y ϭ x2 Ϫ 4x Ϫ 5
72. (4.5) Graph the function r 1x2 ϭ
y
10
x2 Ϫ 4
. Label all
xϪ1
intercepts and asymptotes.
Ϫ10
10 x
Ϫ10
71. (2.3/2.4) State the domain and range of the
functions.
a. y ϭ 12x ϩ 3
b. y ϭ Ϳx ϩ 2Ϳ Ϫ 3
73. (2.6) Suppose the maximum load (in tons) that can
be supported by a cylindrical post varies directly
with its diameter raised to the fourth power and
inversely as the square of its height. A post 8 ft
high and 2 ft in diameter can support 6 tons. How
many tons can be supported by a post 12 ft high
and 3 ft in diameter?
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5.6
Applications from Business, Finance, and Science
LEARNING OBJECTIVES
In Section 5.6 you will see
how we can:
A. Calculate simple interest
and compound interest
Would you pay $750,000 for a home worth only $250,000? Surprisingly, when a conventional mortgage is repaid over 30 years, this is not at all rare. Over time, the accumulated interest on the mortgage is easily more than two or three times the original
value of the house. In this section we explore how interest is paid or charged, and look
at other applications of exponential and logarithmic functions from business, finance,
as well as the physical and social sciences.
B. Calculate interest
compounded continuously
C. Solve applications
of annuities and
amortization
D. Solve applications of
exponential growth and
decay
WORTHY OF NOTE
A. Simple and Compound Interest
Simple interest is an amount of interest that is computed only once during the lifetime
of an investment (or loan). In the world of finance, the initial deposit or base amount is
referred to as the principal p, the interest rate r is given as a percentage and stated as
an annual rate, with the term of the investment or loan most often given as time t in
years. Simple interest is merely an application of the basic percent equation, with the
additional element of time coming into play: interest ϭ principal ϫ rate ϫ time, or
I ϭ prt. To find the total amount A that has accumulated (for deposits) or is due (for
loans) after t years, we merely add the accumulated interest to the initial principal:
A ϭ p ϩ prt.
Simple Interest Formula
If a loan is kept for only a certain
number of months, weeks, or days,
the time t should be stated as a
fractional part of a year so the time
period for the rate (years) matches
the time period over which the loan
is repaid.
If principal p is deposited or borrowed at interest rate r for a period of t years, the
simple interest on this account will be
I ϭ prt
The total amount A accumulated or due after this period will be
A ϭ p ϩ prt or A ϭ p11 ϩ rt2
EXAMPLE 1
ᮣ
Solving an Application of Simple Interest
Many finance companies offer what have become known as PayDay Loans — a
small $50 loan to help people get by until payday, usually no longer than 2 weeks.
If the cost of this service is $12.50, determine the annual rate of interest charged by
these companies.
Solution
ᮣ
The interest charge is $12.50, the initial principal is $50.00, and the time period is
2
1
ϭ 26
2 weeks or 52
of a year. The simple interest formula yields
I ϭ prt
12.50 ϭ 50r a
6.5 ϭ r
simple interest formula
1
b
26
1
for t
substitute $12.50 for I, $50.00 for p, and 26
solve for r
The annual interest rate on these loans is a whopping 650%!
Now try Exercises 7 through 16
5–61
ᮣ
539
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CHAPTER 5 Exponential and Logarithmic Functions
Compound Interest
Many financial institutions pay compound interest on deposits they receive, which is
interest paid on previously accumulated interest. The most common compounding periods are yearly, semiannually (two times per year), quarterly (four times per year),
monthly (12 times per year), and daily (365 times per year). Applications of compound
interest typically involve exponential functions. For convenience, consider $1000 in principal, deposited at 8% for 3 yr. The simple interest calculation shows $240 in interest is
earned and there will be $1240 in the account: A ϭ 1000 31 ϩ 10.082132 4 ϭ $1240. If
the interest is compounded each year 1t ϭ 12 instead of once at the start of the 3-yr period,
the interest calculation shows
A1 ϭ 100011 ϩ 0.082 ϭ 1080 in the account at the end of year 1,
›
A2 ϭ 108011 ϩ 0.082 ϭ 1166.40 in the account at the end of year 2,
›
A3 ϭ 1166.4011 ϩ 0.082 Ϸ 1259.71 in the account at the end of year 3.
The account has earned an additional $19.71 interest. More importantly, notice
that we’re multiplying by 11 ϩ 0.082 each compounding period, meaning results can
be computed more efficiently by simply applying the factor 11 ϩ 0.082 t to the initial
principal p. For example,
A3 ϭ 100011 ϩ 0.082 3 Ϸ $1259.71.
In general, for interest compounded yearly the accumulated value is
A ϭ p11 ϩ r2 t. Notice that solving this equation for p will tell us the amount we need
A
to deposit now, in order to accumulate A dollars in t years: p ϭ 11 ϩ
r2 t . This is called
the present value equation.
Interest Compounded Annually
If a principal p is deposited at interest rate r and compounded yearly for a period of
t yr, the accumulated value is
A ϭ p11 ϩ r2 t
If an accumulated value A is desired after t yr, and the money is deposited at interest
rate r and compounded yearly, the present value is
pϭ
EXAMPLE 2
ᮣ
A
11 ϩ r2 t
Finding the Doubling Time for Interest Compounded Yearly
An initial deposit of $1000 is made into an account paying 6% compounded yearly.
How long will it take for the money to double?
Solution
ᮣ
Using the formula for interest compounded yearly we have
A ϭ p11 ϩ r2 t
2000 ϭ 100011 ϩ 0.062 t
2 ϭ 1.06 t
ln 2 ϭ t ln 1.06
ln 2
ϭt
ln 1.06
11.9 Ϸ t
given
substitute 2000 for A, 1000 for p, and 0.06 for r
isolate variable term
apply base-e logarithms; power property
solve for t
approximate form
The money will double in just under 12 yr.
Now try Exercises 17 through 22
ᮣ