Stewart J. College Algebra: Concepts and Contexts
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394 CHAPTER 4
■
Logarithmic Functions and Exponential Models
Its domain is , and the y-axis is a vertical asymptote of its graph. The graph
for has the shape shown in the margin.
4.2 Laws of Logarithms
Logarithms obey the following Laws of Logarithms:
1.
2.
3.
We can use the Laws of Logarithms to expand or combine logarithmic expressions.
Calculators have a key to evaluate logarithms base 10. To evaluate other
logarithms, we use the Change of Base Formula:
4.3 Logarithmic Scales
When quantities vary over a very large range, we often use a logarithmic scale to
make the range of measurements more manageable. Some examples are as follows:
■
pH scale, for measuring acidity
■
decibel scale, for measuring sound intensity
■
Richter scale, for measuring earthquake intensity
4.4 The Natural Exponential and Logarithmic Functions
The number e is the value that approaches as n becomes large. The value
of e is approximately 2.71828. The natural exponential function is the function
.
The natural logarithm function is the logarithm function with base e; it is de-
noted by ln, so .
The natural exponential and logarithm functions have the following basic
properties:
1. 2. 3. 4.
The natural exponential function can be used to calculate continuously com-
pounded interest. If a principal P is invested at an annual interest rate r compounded
continuously, then the amount to which it grows after t years is given by
The exponential function models exponential growth if or
exponential decay if . Any growth or decay model can be ex-
pressed in terms of e by writing , where .
4.5 Exponential Equations: Getting Information from a Model
An exponential equation is one in which the unknown appears in the exponent of one
or more terms or factors in the equation. For example, is an exponential
equation in the unknown x. To solve an exponential equation, we take logarithms of
both sides and then use the fact that to “bring down the exponent.”log a
x
= x log a
8
x
= 512
r = ln af 1t 2= Ce
rt
f 1x2= Ca
x
r 6 0
r 7 0f 1t2= Ce
rt
A1t 2= Pe
rt
A1t 2
e
ln x
= xln e
x
= xln e = 1ln 1 = 0
ln x = log
e
x
f 1x2= e
x
11 + 1>n 2
n
log
b
x =
log
a
x
log
a
b
LOG
log
a
A
C
= C log
a
A
log
a
1A>B 2= log
a
A - log
a
B
log
a
1AB 2= log
a
A + log
a
B
a 7 1
10, q 2
x
y
0
f(x)=
1
log
a
x
CHAPTER 4
■
Review Exercises 395
A logarithmic equation is one in which the unknown appears inside a loga-
rithm function. For example, is a logarithmic equation. To
solve a logarithmic equation, we first combine all the logarithm terms using the Laws
of Logarithms and then rewrite the equation in exponential form.
Exponential equations are important because they arise when we use exponen-
tial growth or decay models to determine when a certain event has happened or will
happen. For instance, we can determine how old a historical artifact is by using the
decay model for radioactive carbon-14.
4.6 Working with Functions: Composition and Inverse
Given two functions f and , we can make a new function h called the composition
of f with , defined by
The inverse of a function f is a function that “undoes the rule” of f; that is, if we
apply f to the input x and then apply to the result, we get back x again as the fi-
nal output. We have
Only one-to-one functions have inverses. A function is one-to-one if no two dif-
ferent inputs have the same output.
To tell from its graph whether a function is one-to-one we use the Horizontal
Line Test: If no horizontal line intersects the graph more than once, then the func-
tion is one-to-one.
Exponential and logarithmic functions are inverses of each other:
■
If then .
■
If then .g
-1
1x 2= a
x
g1x2= log
a
x
f
-1
1x 2= log
a
xf 1x 2= a
x
f
-1
1f 1x22= xandf 1f
-1
1x 22= x
f
-1
f
-1
h1x 2= f 1g1x 22
g
g
log1x + 1 2- log x = 2
REVIEW EXERCISES
1–2 ■ Find the exact value of each logarithm.
1. (a) (b) (c) (d)
2. (a) (b) (c) (d)
3–4 ■ Complete the table of values for each logarithmic function, and then use your table
to graph the function.
3. 4. g1x 2= log
9
xf 1x 2= log
4
x
log 10
25
log 210log
5
125log
5
1
25
log10.0012log 1000log
4
1
64
log
4
16
CHAPTER
4
SKILLS
x f(x)
0.25
0.5
1
4
8
x
g(x)
1
9
1
3
1
3
9
396 CHAPTER 4
■
Logarithmic Functions and Exponential Models
5–6
■ Use a calculator to find the value of the logarithm correct to two decimals. Use the
Change of Base Formula if necessary.
5. (a) (b) (c) log 94 (d) ln(12.4)
6. (a) (b) (c) (d)
7. An equation is given in logarithmic form. Express it in exponential form.
(a) (b) (c)
8. An equation is given in exponential form. Express it in logarithmic form.
(a) (b) (c)
9–10
■ Use the Laws of Logarithms to expand the given expression.
9. (a) (b)
10. (a) (b)
11–12
■ Use the Laws of Logarithms to combine the given expression.
11. (a) (b)
12. (a) (b)
13–14 ■ Use a graphing calculator and the Change of Base Formula to draw a graph of the
logarithmic function.
13. 14.
15. The hydrogen ion concentration in one cell of a car battery is found to be M.
What is the pH of the acid in this cell? What is it about the pH value that tells you that it
is indeed acidic, not basic?
16. Your upstairs neighbors are having a noisy party, to which you were not invited. You
measure the sound level at /m
2
. Express this in decibels.
17–18
■ Use a graphing calculator to draw a graph of the exponential function.
17. 18.
19–20
■ A growth or decay model is given. Express the model in terms of the base e.
19. 20.
21–22
■ Find an exponential model of the form for the population that is
described (where t is measured in hours).
21. The initial population is 1500, and it triples every hour.
22. The initial population is 600, and it increases by 50% every hour.
23–30
■ Solve the equation for the unknown x.
23. 24.
25. 26.
27. 28.
29. 30.
31–34
■ Two functions f and are given.
(a) Find algebraic expressions for the functions and .
(b) Evaluate and .g1f 1- 322f 1g1222
g1f 1x22f 1g1x 22
g
ln1x + 32= 5 + ln xlog
4
x = 2 + log
4
12x - 32
log
2
x = 3 + log
2
1x - 12log1x + 12- log x = 1
17e
3x
= 125
x - 1
= 47
7
#
2
3x
= 563
2x + 1
= 27
f 1t 2= Ce
rt
g1t 2= 12A
1
4
B
t
f 1t 2= 2400
#
3
t
g1x 2= e
-0.5x
f 1x 2=
1
2
e
0.3x
55 W
7.3 * 10
-2
g1x 2= log
15
xf 1x 2= log
6
x
log1a - 22+ 3 log alog
5
1b + 12- 4 log
5
1b + 32
ln1x + 12- 2 ln y3 log
2
X + log
2
Y
ln1xyz
2
2log
4
a
3
b
log
5x
y
2
log
2
13ab 2
4
y
= 73
x
= 169
1>2
= 3
ln y = 4log
2
x = 3log
5
25 = 2
ln 50log 50log
2
10.33 2log
8
67
log
2
1000log
6
27
CHAPTER 4
■
Review Exercises 397
31. 32.
33. 34.
35–38
■ Find the inverse function of f.
35. 36.
37. 38.
39–40
■ Use a graphing calculator to draw the graph of f. Then use the graph to determine
whether the function is one-to-one.
39. 40. f 1x 2= x
3
- 4xf 1x2= x
3
+ 4x
f 1x 2= log a
x
6
bf 1x 2= 8e
x
f 1x 2=
2
x + 1
f 1x 2= 3x - 7
f 1x 2= log x,
g1x 2= 10
3x
f 1x 2= 2x
2
,g1x 2= x
3
f 1x 2= x
2
,g1x 2= 1 - 2xf 1x 2= 2x - 4,g1x2= 3x + 1
These exercises test your understanding by combining ideas from several sections in a
single problem.
41. Drug Metabolism Herman is prescribed a blood pressure medication, of which he
must take an 80-milligram dose at the same time every morning. Research has shown
that in a man of Herman’s size, 10% of the drug is eliminated from the body in each
4-hour period. In this problem we consider what happens to his first dose of the drug.
(a) Find an exponential function of the form that models the amount
of the drug in Herman’s body x 4-hour time periods after he ingests it.
(b) Rewrite your model from part (a) as a function of the form , where t is
the number of hours since Herman ingests the drug. What is the 1-hour decay
factor? The 1-hour decay rate?
(c) For best therapeutic effect, the drug level should always be above 50 milligrams.
Use a graphing calculator to graph the function N over the 24-hour period between
doses. Does it appear that Herman always has 50 milligrams or more of the drug in
his body? If not, determine the time at which the drug falls to a level of 50
milligrams. (Find your solution both graphically and algebraically.)
(d) Express the model using the natural exponential function: . What is the
instantaneous decay rate?
(e) How much of the drug is still in Herman’s system after 24 hours (when he is ready
to take his next dose)?
(f) If we want to construct a model for the second day’s drug levels, what change will
we have to make to the function N from part (b)? (Keep in mind that on the second
day Herman starts out with some of the drug already in his system, from the pre-
ceding day’s pill.) Find a function that models the drug levels on the second
day.
(g) Graph the function you found in part (f). Will the drug level fall below 50
milligrams on the second day?
42. Dating an Ancient Relic During the times of the Roman Empire much of present-
day northern Germany was inhabited by Germanic tribes that were never conquered by
Rome. It is not uncommon to find relics from these ancient peoples in bogs and lake
bottoms.
While digging a foundation for his new house in Mecklenburg-Vorpommern,
Dietmar discovers a well-preserved rowboat carved out of beechwood. A test at the
local university reveals that the wood has 73% of the carbon-14 that exists in current
plant life.
(a) Use the fact that carbon-14 has a half-life of 5370 years to find a model of the form
for the amount of carbon-14 remaining after t years from a
sample with initial mass C.
A1t 2A1t 2= CA
1
2
B
t>h
S1t 2
N1t 2= Ce
rt
N1t 2= Cb
t
M1x 2M1x 2= Ca
x
CONNECTING
THE CONCEPTS
398 CHAPTER 4
■
Logarithmic Functions and Exponential Models
(b) Convert your model from part (a) to a function of the form . The value
of r should be negative. What does this mean?
(c) What percentage of the original carbon-14 should remain in a sample that is 2000
years old?
(d) Assume that the initial amount C is 100 grams in the function of part (b). Find a
formula for the function . What does represent (if x is a mass measured
in grams)?
(e) How old is the rowboat that Dietmar dug up? Could it be a relic from the ancient
Germanic tribes?
43. pH of Soil The acidity of soil has a major effect on the types of plants that can be grown
in it. Beans do not grow well in soil that is strongly acidic, with a pH of less than 5.3.
(a) A chemist tests the soil in her back yard and finds that its hydrogen ion
concentration is M. What is the pH of the soil? Is it suitable for
growing beans?
(b) The chemist amends the soil to change its pH to 5.9. What is the hydrogen ion
concentration of the amended soil?
44. Earthquake Magnitude Seismographs were not in common use until the 20th
century, so it is difficult to gauge the strength of earthquakes that occurred earlier than
that. Nevertheless, scientists estimate that the Lisbon earthquake of November 1, 1755,
which completely destroyed the Portuguese capital and caused a massive tsunami, had a
magnitude of 9.0 on the Richter scale. A prior earthquake in Lisbon in 1714 had only
2% of the intensity of the 1755 event. What was the magnitude of the 1714 earthquake
on the Richter scale?
45. Compound Interest Jennifer invests $250,000 in a three-year CD that pays 4.25%
annual interest, compounded continuously.
(a) What is the instantaneous growth rate of her CD?
(b) What will the value of the CD be when it matures after three years?
(c) If the CD had compounded the interest annually instead of continuously, how much
would it have been worth at maturity?
46. Compound Interest Zacky has $5000 that he wants to invest in a secure account. His
bank offers him the choice of two different guaranteed interest savings accounts. How
long will it take for his deposit to grow to $7000 in each of these accounts?
(a) The account pays 3.75% annual interest, compounded monthly.
(b) The account pays 3.62% annual interest, compounded continuously.
47. Size of a Turtle Most organisms have genetic limiters on their growth, so they
increase in size only when they are young. Dogs, for example, reach their maximum
size in the first 15 months of life and stop growing after that. However, some reptiles
grow throughout their life, a phenomenon known as attenuated growth.
Hortense is a loggerhead turtle in the Norristown zoo. Her weight (in pounds)
t years since she was acquired by the zoo in 2000 is modeled by the function
(a) How many years does it take for Hortense to double her weight?
(b) Express Hortense’s weight as a function of the form . What is the
instantaneous growth rate of her weight?
(c) When will Hortense weigh 150 pounds?
(d) Loggerhead turtles sometimes live 150 years or more. If Hortense lives to the year
2100, how much will she weigh then, according to the model? Does this seem
reasonable? What kind of model might be a better long-term predictor of her
weight?
W1t 2= Ce
rt
W1t 2= 23
#
2
t>6
3H
+
4= 3.7 * 10
-7
A
-1
1x 2A
-1
A1t 2= Ce
rt
CONTEXTS
CHAPTER 4
■
Review Exercises 399
48. Time of Death The deer-hunting season in Wappaloosa County opens at noon on
October 15. A game warden interviews a hunter who has a deer carcass in his pickup
truck at 3:00
P.M. on opening day. Normal body temperature for deer is , and the
air temperature on this day is . Under these circumstances, Newton’s Law of
Cooling indicates that t hours after death, a deer will have cooled to T degrees, where
(a) The warden measures the deer’s temperature and finds it to be . How long ago
did the deer die?
(b) Was the deer killed before the official season opening, or was it shot legally?
49. Multiple Coupons A grocery store distributes gasoline coupons of two types to its
customers, based on the amount of their grocery purchase. The Type A coupon provides
a 5% discount, and the Type B coupon provides a $2.00 discount on the total gasoline
purchase. If you buy a tankful of gasoline priced at x dollars, let be your final cost
when using a Type A coupon, and let be your final cost when using a Type B
coupon.
(a) Find formulas for the functions f and .
(b) The gasoline retailer permits the use of one of each type of coupon on a single
purchase. What do the functions and represent?
(c) Find formulas for both and .
(d) Find and . Does the order in which the coupons are applied to
your purchase make a difference?
50. Cost of Services A house cleaner decides to charge his customers using an
exponential scale, to encourage them to maintain their homes more neatly between
cleanings. For a cleaning that takes him x hours, he charges dollars.
(a) How much would a cleaning cost if it takes 1 hour? 3 hours?
(b) Find a formula for . What does this function represent?
(c) A customer receives a bill for $388.96. How long did it take to clean this person’s
home?
C
-1
1x 2
C1x 2= 2012.1 2
x
g1f 14022f 1g14022
g1f 1x22f 1g1x 22
g1f 1x22f 1g1x 22
g
g1x 2
f 1x 2
85°F
T = 68 + 1102 - 682e
-0.095t
68°F
102°F
400 CHAPTER 4
■
Logarithmic Functions and Exponential Models
TEST
1. Find the exact value of the logarithm.
(a) (b) (c)
2. Let .
(a) Use the Change of Base Formula to complete the table of values below, correct to
two decimal places.
f 1x 2= log
3
x
log 10
100
log
9
27log
4
A
1
16
B
CHAPTER
4
(b) Use the table of values to sketch a graph of f.
3. Use the Laws of Logarithms to perform the indicated operation on the expression.
(a) Expand:
(b) Combine as a single logarithm:
4. If the hydrogen ion concentration in a solution is moles/liter (M), then its pH is
.
(a) Find the pH of a yogurt dip whose hydrogen ion concentration is M.
(b) Find the hydrogen ion concentration of a glass of lemonade with a pH of 4.3.
5. Karl-Heinz is trying to decide where to invest his tax refund of $2325. Calculate the
amount his investment will grow to after 3 years under the following CD choices
offered by competing banks. Which is the better investment?
(a) 3% annual interest, compounded daily
(b) 2.95% annual interest, compounded continuously
6. Rhodium-106 has a half-life of 30 seconds, so its decay is modeled by the equation
, where is the amount remaining after t seconds of a sample that
initially weighted C grams.
(a) Express the model in the form . What is the instantaneous decay rate?
(b) How long will it take for a 1000-gram sample of rhodium-106 to decay to 1 gram?
7. Solve the equation for the unknown x. Express your answer correct to two decimal
places.
(a) (b)
8. An online book club charges its members a shipping fee of $2.95 for the first book
ordered plus $1.00 for each additional book.
(a) Let be the shipping fee on an order of x books. Find a formula for the
function S.
(b) Find a formula for . What does represent?
(c) If a customer’s order has a shipping fee of $12.95, how many books were ordered?
9. Let , and let .
(a) Find formulas for the compositions and .
(b) Evaluate and .g1f 1022f 1g1022
g1f 1x22f 1g1x 22
g1x 2= log1x - 1 2f 1x2
= 10
x + 2
S
-1
1x 2S
-1
S1x 2
2 log
4
x - log
4
1x
2
+ x - 42= 07
#
5
3x
= 105
A1t 2= Ce
rt
A1t 2A1t2= CA
1
2
B
t>30
4.3 * 10
-7
- log3H
+
4
3H
+
4
ln 5 + 2 ln x - ln1x
3
+ 72
log a
5x
4
6x - 11
b
x 0.1 0.5 1 3 5
f 1x 2
0 folds 1 fold 2 folds
Super Origami
OBJECTIVE To explore how logarithms are used to solve exponential equations.
Origami is the traditional Japanese art of paper folding. The Japanese word origami
literally means “paper folding” (from oru, meaning “folding,” and kami, meaning
“paper”). The goal of this art is to create an illustration of an object by folding a sheet
of paper. Most of us have learned how to make paper airplanes or simple paper boats.
One of the most common origami constructions is the crane shown here. But origami
masters can fold paper into creative forms, including animals, birds, flowers, build-
ings, and towers. In this exploration we investigate the simplest type of origami: just
folding a sheet of paper over and over again.
I. How to Make a Paper Tower
Take an ordinary sheet of paper and fold it in half. Now fold it in half again. Keep
folding the paper in half. How many times are you able to fold it?
EXPLORATIONS
■
EXPLORATIONS
■
EXPLORATIONS
■
EXPLORATIONS
1
Slash331/Shutterstock.com 2009
1. (a) When you fold the sheet of paper once, the result is a stack with two
layers of paper; folding it twice results in a four-layer stack, and so on.
Complete the table for the number of layers after x folds.
Number of folds x 0 1 2 3 4 5
Layers f 1x2
1
(b) The number of layers grows exponentially. What is the growth factor?
What is the initial value? Find an exponential function that
models the number of layers after x folds.
x
2. Suppose the sheet of paper we started with is 1/1000 of an inch thick.
(a) How thick is the stack of paper after 10 folds?
Number of layers: _______
Thickness in inches: _______
(b) How thick is the stack of paper after 20 folds?
Number of layers: _______
Thickness in inches: _______
Thickness in feet: _______
f 1x2=
ⵧ
ⴢ
ⵧ
f 1x 2= Ca
x
2
2
EXPLORATIONS 401
A still from the film Powers of Ten
(c) How thick is the stack of paper after 50 folds?
Number of layers: _______
Thickness in miles: _______
II. How High a Tower?
How many times do you need to fold the paper to make a tower as tall as you are?
Or as tall as the Empire State Building? To answer these questions, let’s start with a
sheet of paper that is 1/1000 of an inch thick.
1. How many sheets of paper are needed to form a stack of paper with the given
height?
1 inch: _______
1 foot: _______
1 mile: _______
2. How many times do you need to fold the paper to get a stack of paper as tall
as you are? Answer this question in the following steps:
Express your height in inches: h ⫽ _______
Express your height in sheets of paper: N ⫽ _______
Solve the equation for x. Explain why your answer is the number of
folds you need to get a stack your height. Can you actually fold the paper that
many times?
3. How many times do we need to fold the paper to get a stack with the
following heights?
(a) The height of the Empire State Building (1250 feet)
(b) The height of Mount Everest (29,029 feet)
(c) The distance to the moon (240,000 miles)
(d) The distance to the sun (93,000,000 miles)
2
x
= N
Orders of Magnitude
OBJECTIVE To compare sizes of objects using logarithms or orders of magnitude.
Our universe presents us with enormous disparities in size—from incredibly tiny
subatomic particles to unbelievably huge galaxies. How can we meaningfully com-
pare these sizes? The 1977 short documentary film Powers of Ten, created by Ray
and Charles Eames, uses zoom photography to visually compare the relative sizes of
different objects in the universe. (This delightful film is available for viewing on sev-
eral Internet sites.) We also encounter disparities in size in our everyday life—for ex-
ample, our own income in comparison to trillion dollar national budgets (or deficits).
But to bring these disparities in size down to earth, let’s just try to fathom the wealth
of the billionaire next door.
2
2
EXPLORATIONS
■
EXPLORATIONS
■
EXPLORATIONS
■
EXPLORATIONS
402 CHAPTER 4
0
1 million dollars
The millionaire measurin
g
tape
1 2 3 4 5 6
EXPLORATIONS
■
EXPLORATIONS
■
EXPLORATIONS
■
EXPLORATIONS
EXPLORATIONS 403
I. How Rich Is a Billionaire?
A millionaire is rich; he or she must have at least a million dollars in the bank. A bil-
lionaire is richer. Let’s compare the relative wealth of a millionaire and a billionaire
graphically and numerically.
1. Let’s use the millionaire measuring tape to measure wealth. Each mark on
the tape represents one million dollars. So the mark at 1 represents one million
dollars, the mark at 2 represents two million dollars, and so on.
(a) At what number on the tape is the one billion dollar mark? _______
(b) If the marks on the tape are one inch apart, how many inches to the right
of zero is the billion dollar mark? _______
(c) How many feet to the right of zero is the billion dollar mark? _______
2. The billion dollar mark on the millionaire measuring tape in Question 1 is way
off the page. So let’s use the billionaire ruler, on which the mark at 1 repre-
sents one billion dollars.
(a) Where is the 100 million dollar mark on the ruler?
(b) Place marks on the ruler at 1 million and 2 million dollars. Is it easy to
visually distinguish the two marks?
(c) Place marks at one thousand and ten thousand dollars on the ruler.
3. The “millionaire measuring tape” and the “billionaire ruler” are not so easy
to use.
■
The millionaire measuring tape is just too long to handle easily.
■
The billionaire ruler can’t be easily used to measure small amounts.
So let’s try the “logarithm ruler.” Each mark on the logarithm ruler is the
logarithm (base 10) of the number it represents. So the mark at 1 represents
dollars, the mark at 2 represents dollars, and so on. Note that each
mark on this ruler represents a number that is ten times the number of the
preceding mark.
10
2
10
1
0
1 billion dollars
The billlionaire ruler
1