314
Chapter 4
Exponential and Logarithmic Functions
years. Let’s see . . . 12% of the $5,000 is $600 and $5,600 divided by 36 months
is $155.56, but I’ll pay you $160 per month for the trouble of carrying the loan.
Is it a deal?”
(a) If this deal sounds fair to you, I have a perfectly lovely bridge I think you
should consider as your next purchase. If not, explain why the deal is fishy
and compute a fair monthly payment (assuming you still plan to amortize the
debt of $5,000 over 3 years).
(b) Read an article on truth in lending and think up some examples of plausible
yet shady deals, such as the proposed used-car transaction in this problem.
46. Program a computer or use a calculator to evaluate 1 1
n
n
for n 1,000,
2,000, . . . , 50,000.
1 n
for n 1,000,
n
2,000, . . . 50,000. On the basis of these calculations, what can you conjecture
1 n
about the behavior of 1 as n decreases without bound?
n
47. Program a computer or use a calculator to evaluate 1 48. Program a computer or use a calculator to estimate lim
nfi .
2 2n5 .
1
3
n
2n
n/3
49. Program a computer or use a calculator to estimate lim
nfi 2
Logarithmic
Functions
Suppose you invest $1,000 at 8% compounded continuously and wish to know how
much time must pass for your investment to double in value to $2,000. According to
the formula derived in Section 1, the value of your account after t years will be
1000e0.08t, so to find the doubling time for your account, you must solve for t in the
equation
1,000e0.08t 2,000
or, by dividing both sides by 1,000,
e0.08t 2
We will answer the question about doubling time in Example 2.9. Solving an
exponential equation such as this involves using logarithms, which reverse the process
of exponentiation. Logarithms play an important role in a variety of applications, such
as measuring the capacity of a transmission channel and in the famous Richter scale
for measuring earthquake intensity. In this section, we examine the basic properties
of logarithmic functions and a few applications. We begin with a definition.
Chapter 4 ■ Section 2
Store y ex into Y1, using a
bold graphing style, and y x
into Y2. Since y ln x is a representation of ey x, we can
display its graph by drawing it
as the inverse relation of y ex.
Use a decimal window.
Logarithmic Functions
315
Logarithmic Functions ■ The logarithm of x to the base b is the
number y such that by x and we write y logb x. In the case where b e,
the number y such that ey x is called the natural logarithm of x and is written y ln x (read as “el en x”); that is,
y ln x
if and only if
ey x
In all cases, the logarithm is defined only for x 0.
EXAMPLE 2.1
Evaluate
(a) log10 1,000
(b) log2 32
(c) log5
1251 .
Solution
(a) log10 1,000 3 since 103 1,000.
(b) log2 32 5 since 25 32.
1
1
(c) log5
3 since 53 .
125
125
EXAMPLE 2.2
Solve each of the following equations for x:
1
(a) log4 x (b) log64 16 x
(c) logx 27 3
2
Solution
1
is equivalent to x 41/2 2.
2
(b) log64 16 x means 16 64x or 24 (26)x 26x and by equating exponents on
2
each side of this last equation, we get 4 6x so x .
3
(c) logx 27 3 means x3 27, so x 271/3 3.
(a) By definition, log4 x 316
Chapter 4
Exponential and Logarithmic Functions
Most of our work will be with natural logarithms. To evaluate ln a for a particular number a, you use the “LN” key on your calculator. For example, to find
ln 2.714, you would press the LN key, then enter the number 2.714 to get
ln 2.714 0.9984
to four decimal places. Here is an example illustrating the computation of natural
logarithms.
EXAMPLE 2.3
Find
(a) ln e
(b) ln 1
(c) ln e
(d) ln 2
Solution
(a) According to the definition, ln e is the unique number b such that e eb. Clearly
this number is b 1. Hence, ln e 1.
(b) ln 1 is the unique number b such that 1 eb. Since e0 1, it follows that
ln 1 0.
1
(c) ln e ln e1/2 is the unique number b such that e1/2 eb; that is, b . Hence,
2
1
ln e .
2
(d) ln 2 is the unique number b such that 2 eb. The value of this number is not
obvious, and you will have to use your calculator to find that ln 2 0.69315.
THE RELATIONSHIP
ex AND ln x
BETWEEN
The next example establishes two important identities that show how logarithmic and
exponential functions have a certain “neutralizing” effect on each other.
EXAMPLE 2.4
Simplify the following expressions:
(a) eln x (for x 0)
(b) ln ex
Solution
(a) According to the definition, ln x is the unique number b for which x eb. Hence,
eln x eb x.
(b) Similarly, ln ex is the unique number b for which ex eb. Clearly, this number b
is x itself. Hence, ln ex x.
Chapter 4 ■ Section 2
Logarithmic Functions
317
The two identities derived in Example 2.4 show that the composite functions
ln ex and eln x leave the variable x unchanged. In general, two functions f and g for
which f(g(x)) x and g( f(x)) x are said to be inverses of one another. Thus, the
exponential function y ex and the logarithmic function y ln x are inverses.
The Inverse Relationship between ex and ln x
ln x
e
x
for x 0
ln e x
x
and
■
for all x
The next example illustrates how you can use the inverse relationship between
ex and ln x to solve equations.
EXAMPLE 2.5
Solve the equation 3 ex ln (x2 1) by placing the left
side of the equation into Y1 and
the right side into Y2. Use a
standard window (zoom 6) to
find the x values of the intersection points.
Solve each of the following equations for x:
(a) 3 e20x
(b) 2 ln x 1
Solution
(a) Take the natural logarithm of each side of the equation to get
ln 3 ln e20x
ln 3 20x
or
Solve for x (using a calculator) to find ln 3:
x
ln 3 1.0986
0.0549
20
20
(b) First isolate ln x on the left side of the equation by dividing both sides by 2:
ln x 1
2
Then apply the exponential function to both sides of the equation to get
eln x e1/2
THE GRAPH OF ln x
or
x e1/2 e 1.6487
There is an easy way to obtain the graph of the logarithmic function y ln x from
the graph of the exponential function y ex. The method is based on the geometric
fact that the point (b, a) is the reflection of the point (a, b) across the line y x. This
fact is illustrated in Figure 4.5, and a proof is outlined in Problem 54.
318
y
y=x
(a, b)
b
a
(b, a)
x
a b
Chapter 4
Exponential and Logarithmic Functions
The idea is that since y ln x means that x ey, the graph of y ln x is the
same as the graph of y ex with the roles of x and y reversed. More precisely, suppose (a, b) is a point on the curve y ln x. Then b ln a, or equivalently, a eb.
Hence, the reflected point (b, a) can be written as (b, eb), which is a point on the
curve y ex. Conversely, if (a, b) is on the curve y ex, it follows that b ea and
a ln b, so the reflected point is (b, a) (b, ln b), which lies on the curve y ln x. To summarize:
Relationship between the Graphs of y 5 ln x and y 5
ex ■ The graph of y ln x is obtained by reflecting the graph of y ex
FIGURE 4.5 Reflection of points
across the line y x.
across the line y x.
The graph of the exponential function y ex, the line y x, and the graph of
the logarithmic function y ln x are sketched in Figure 4.6. Notice that the natural
logarithm function has the following properties:
1. ln x is defined only for x 0.
2. ln 1 0 and ln e 1.
3. ln x increases without bound as x increases without bound and decreases without
bound as x approaches zero from the right; that is,
lim ln x lim ln x and
xfi xfi 0 y
y=x
y = ex
(1, e)
(0, 1)
(e, 1)
y = 1n x
x
(1, 0) e
FIGURE 4.6 The graph of y ln x is the reflection of the graph of y ex across the line y x.
Chapter 4 ■ Section 2
PROPERTIES OF THE NATURAL
LOGARITHM
Logarithmic Functions
319
The laws of exponents can be used to derive the following important properties of
logarithms.
Properties of Logarithms
ln u ln v if and only if u v
The equality rule:
The product rule: ln uv ln u ln v
ln ur r ln u
The power rule:
The quotient rule:
y
ln
u
ln u ln v
v
y = ln x
The equality rule follows from the fact that the logarithm curve is always rising
(see Figure 4.7). To prove the product rule, let m logb u and n logb v so
ln b
ln a
x
a
u bm
b
v bn
and
Then
logb uv logb bmbn logb bmn
m n logb u logb v
FIGURE 4.7 Equality rule: Since
the graph of y ln x always
rises, a b implies ln a ln b.
Proofs of the power rule and quotient rule are outlined in the exercises (Problem 52).
EXAMPLE 2.6
(a) Find ln ab if ln a 3 and ln b 7.
1
(b) Show that ln ln x.
x
(c) Find x if 2x e3.
Solution
(a) ln ab ln (ab)1/2 (b) ln
1
1
1
ln ab (ln a ln b) (3 7) 5
2
2
2
1
ln 1 ln x 0 ln x ln x
x
(c) Take the natural logarithm of each side of the equation 2x e3 and solve for x
to get
x ln 2 3
or
x
3
4.33
ln 2
320
Chapter 4
Exponential and Logarithmic Functions
You may have noticed that your calculator has a “Yx” key for handling general
exponential functions, but has only the “LN” key for handling logarithms. This is no
problem, thanks to the following conversion formula:
Conversion Formula for Logarithms
■
If b (b 0, b 1) is
any base other than e, then
logb x ln x
ln b
To prove the conversion formula, note that y logb x is equivalent to x by.
Then by taking the natural logarithm on both sides of this equation and applying the
power rule for logarithms, we get
x by
ln x ln by y ln b
so
logb x y ln x
ln b
EXAMPLE 2.7
Find log5 3.172.
Solution
Using the conversion formula, you find
log5 3.172 Put the function y 10x into Y1
and graph using a bold graphing
style. Then put y x into Y2,
y ln x into Y3, and y log x into Y4. What can you
conclude from this series of
graphs?
ln 3.172
1.1544
0.7173
ln 5
1.6094
EXAMPLE 2.8
If log10 x 3, what is ln x?
Solution
Using the conversion formula, we find that
3 log10 x ln x
ln 10
Chapter 4 ■ Section 2
Logarithmic Functions
321
Solving the equation for ln x, we get
Note
ln x 3 ln 10 6.9078
DOUBLING TIME
Use the equation solver of your
graphing calculator with the
equation F P*e^(R*T) 0 to
determine how long it will take
for $2,500 to double at 8.5%
compounded continuously.
Some calculators have a “LOG” key for logarithms to base 10.
In the introductory paragraph at the beginning of this section, you were asked how
long it would take for a particular investment to double in value. This question is
answered in the following example.
EXAMPLE 2.9
If $1,000 is invested at 8% annual interest, compounded continuously, how long will
it take for the investment to double? Would the doubling time change if the principal
were something other than $1,000?
Solution
With a principal of $1,000, the balance after t years is B(t) 1,000e0.08t, so the investment doubles when B(t) $2,000; that is, when
2,000 1,000e0.08t
Dividing by 1,000 and taking the natural logarithm on each side of the equation, we
get
2 e0.08t
ln 2 0.08t
ln 2
t
8.66 years
0.08
If the principal had been P0 dollars instead of $1,000, the doubling time would
satisfy
2P0 P0e0.08t
2 e0.08t
which is exactly the same equation we had with P0 $1,000, so once again, the doubling time is 8.66 years.
322
Chapter 4
Exponential and Logarithmic Functions
The situation illustrated in Example 2.9 applies to any quantity Q(t) Q0ekt that
grows exponentially with time t. In particular, since at time t 0, we have Q(0) Q0e0 Q0, the quantity doubles when
2Q0 Q0ekt
2 ekt
ln 2 kt
ln 2
t
k
To summarize:
A quantity Q(t) Q0ekt (k 0) undergoing exponential growth doubles when
Doubling Time
■
t
HALF-LIFE
ln 2
k
It has been experimentally determined that most radioactive substances decay exponentially, so that the amount of a sample of initial size Q0 that is present after t years
is given by a function of the form Q(t) Q0ekt. The positive constant k measures
the rate of decay, but this rate is usually given by specifying the amount of time t
required for half a given sample to decay. This time is called the half-life of the
radioactive substance. Example 2.10 shows how half-life is related to k.
EXAMPLE 2.10
Show that a radioactive substance that decays according to the formula Q(t) Q0ekt
ln 2
has half-life t .
k
Solution
The goal is to find the value of t for which Q( t ) 1
Q0; that is,
2
1
Q0 Q0ekt
2
Divide by Q0 and take the natural logarithm of each side to get
ln
1
kt
2
Chapter 4 ■ Section 2
Logarithmic Functions
ln
Thus, the half-life is
t
k
1
2
323
ln 2
k
as required.
CARBON DATING
In 1960, W. F. Libby won a Nobel prize for his discovery of carbon dating, a technique for determining the age of certain fossils and artifacts. Here is an outline of the
technique.*
The carbon dioxide in the air contains the radioactive isotope 14C (“carbon 14”)
as well as the stable isotope 12C (“carbon 12”). Living plants absorb carbon dioxide
from the air, which means that the ratio of 14C to 12C in a living plant (or in an animal that eats plants) is the same as that in the air itself. When a plant or an animal
dies, the absorption of carbon dioxide ceases. The 12C already in the plant or animal
remains the same as at the time of death while the 14C decays, and the ratio of 14C to
12
C decreases exponentially. It is reasonable to assume that the ratio R0 of 14C to 12C
in the atmosphere is the same today as it was in the past, so that the ratio of 14C to
12
C in a sample (e.g., a fossil or an artifact) is given by a function of the form
R(t) R0ekt. The half-life of 14C is 5,730 years. By comparing R(t) to R0, archaeologists can estimate the age of the sample. Example 2.11 illustrates the dating procedure.
EXAMPLE 2.11
An archaeologist has found a fossil in which the ratio of
14
C to
12
C is
1
the ratio
5
found in the atmosphere. Approximately how old is the fossil?
Solution
The age of the fossil is the value of t for which R(t) 1
R0; that is, for which
5
1
R0 R0ekt
5
Dividing by R0 and taking logarithms, you find that
* For instance, see Raymond J. Cannon, “Exponential Growth and Decay,” UMAP Modules 1977: Tools
for Teaching, Consortium for Mathematics and Its Applications, Inc., Lexington, MA, 1978. More
advanced dating procedures are discussed in Paul J. Campbell. “How Old Is the Earth?” UMAP
Modules 1992: Tools for Teaching, Consortium for Mathematics and Its Applications, Inc., Lexington,
MA, 1993.
324
Chapter 4
Exponential and Logarithmic Functions
1
ekt
5
1
ln kt
5
1
ln
5 ln 5
t
k
k
and
In Example 2.10, you found that the half-life t satisfies t ln 2
, and since
k
14
C has
half-life t 5,730 years, you have
k
ln 2
ln 2
0.000121
5,730
t
Therefore, the age of the fossil is
t
ln 5
ln 5
13,300
k
0.000121
That is, the fossil is approximately 13,300 years old.
EXPONENTIAL CURVE FITTING
In our final example, you will see how to use logarithms to fit an exponential function to a set of data.
EXAMPLE 2.12
Refer to Example 2.12. Place
0 Q A*e^(K*X) into the
equation editor of your graphing calculator. Find the distance
from the center of the city if
Q 13,500 people per square
mile. Recall from the example
that the density at city center is
15,000 and the density 10 miles
from city center is 9,000 people
per square mile.
The population density x miles from the center of a city is given by a function of the
form Q(x) Aekx. Find this function if it is known that the population density at
the center of the city is 15,000 people per square mile and the density 10 miles from
the center is 9,000 people per square mile.
Solution
For simplicity, express the density in units of 1,000 people per square mile. The fact
that Q(0) 15 tells you that A 15. The fact that Q(10) 9 means that
9 15e10k
or
3
e10k
5
Taking the logarithm of each side of this equation, you get
ln
3
10k
5
or
k
ln 3/5
0.051
10
Chapter 4 ■ Section 2
Logarithmic Functions
325
Hence the exponential function for the population density is (approximately) Q(x) 15e0.051x.
P . R . O . B . L . E . M . S
4.2
In Problems 1 and 2, use your calculator to find the indicated natural logarithms.
1. Find ln 1, ln 2, ln e, ln 5, ln
1
, and ln e2. What happens if you try to find ln 0 or
5
ln (2)? Why?
2. Find ln 7, ln
1
1
5
, ln e3, ln 2.1 , and ln e.
3
e
In Problems 3 through 8, evaluate the given expression using properties of the
logarithm.
3. ln e3
4. ln e
ln 5
5. e
6. e2 ln 3
7. e3 ln 22 ln 5
8. ln
e3e
e1/3
In Problems 9 through 20, solve the given equation for x.
9. 2 e0.06x
11. 3 2 5e4x
13. ln x t
C
50
1
10. Q0 Q0e1.2x
2
12. 2 ln x b
14. 5 3 ln x 1
ln x
2
1
15. ln x (ln 16 2 ln 2)
3
16. ln x 2(ln 3 ln 5)
17. 3x e2
18. ak ekx
19. ax1 b
20. xln x e
21. If log2 x 5, what is ln x?
22. If log10 x 3, what is ln x?
23. If log5 (2x) 7, what is ln x?
24. If log3 (x 5) 2, what is ln x?
326
Chapter 4
Exponential and Logarithmic Functions
25. Find ln
26. Find
1
if ln a 2 and ln b 3.
ab3
1
b
ln
a
c
a
if ln b 6 and ln c 2.
COMPOUND INTEREST
27. How quickly will money double if it is invested at an annual interest rate of 6%
compounded continuously?
COMPOUND INTEREST
28. How quickly will money double if it is invested at an annual interest rate of 7%
and interest is compounded continuously?
COMPOUND INTEREST
29. Money deposited in a certain bank doubles every 13 years. The bank compounds
interest continuously. What annual interest rate does the bank offer?
RADIOACTIVE DECAY
30. The amount of a certain radioactive substance remaining after t years is given by
a function of the form Q(t) Q0e0.003t. Find the half-life of the substance.
RADIOACTIVE DECAY
31. Radium decays exponentially. Its half-life is 1,690 years. How long will it take
for a 50-gram sample of radium to be reduced to 5 grams?
ADVERTISING
32. The mathematics editor at a major publishing house estimates that if x thousand
complimentary copies are distributed to instructors, the first-year sales of a new
mathematics text will be approximately f(x) 20 15e0.02x thousand copies.
According to this estimate, approximately how many complimentary copies should
the editor send out to generate first-year sales of 12,000 copies?
GROWTH OF BACTERIA
33. A medical student studying the growth of bacteria in a certain culture has compiled the following data:
Number of minutes
0
20
Number of bacteria
6,000
9,000
Use these data to find an exponential function of the form Q(t) Q0ekt expressing the number of bacteria in the culture as a function of time.
GROSS DOMESTIC PRODUCT
34. An economist has compiled the following data on the gross domestic product
(GDP) of a certain country:
Year
1990
2000
GDP (in billions)
100
180
Use these data to predict the GDP in the year 2010 if the GDP is growing:
(a) Linearly
(b) Exponentially
WORKER EFFICIENCY
35. An efficiency expert hired by a manufacturing firm has compiled the following
data relating workers’ output to their experience:
Chapter 4 ■ Section 2
Experience (months)
Output (units per hour)
Logarithmic Functions
0
6
300
410
327
The expert believes that the output Q is related to experience t by a function of
the form Q(t) 500 Aekt. Find the function of this form that fits the data.
ARCHAEOLOGY
36. An archaeologist has found a fossil in which the ratio of
14
C to
12
C is
1
the ratio
3
found in the atmosphere. Approximately how old is the fossil?
ARCHAEOLOGY
37. Tests of an artifact discovered at the Debert site in Nova Scotia show that 28%
of the original 14C is still present. Approximately how old is the artifact?
ARCHAEOLOGY
38. The Dead Sea Scrolls were written on parchment in about 100 B.C. What percentage of the original 14C in the parchment remained when the scrolls were discovered in 1947?
ART FORGERY
39. A forged painting allegedly painted by Rembrandt in 1640 is found to have 99.7%
of its original 14C. When was it actually painted? What percentage of the original 14C should remain if it were legitimate?
DEMOGRAPHICS
40. The world’s population grows at the rate of approximately 2% per year. If it is
assumed that the population growth is exponential, then the population t years
from now will be given by a function of the form P(t) P0e0.02t, where P0 is the
current population. (This formula is derived in Chapter 5.) Assuming that this
model of population growth is correct, how long will it take for the world’s population to double?
RADIOLOGY
41. Radioactive iodine 133I has a half-life of 20.9 hours. If injected into the bloodstream, the iodine accumulates in the thyroid gland.
(a) After 24 hours, a medical technician scans a patient’s thyroid gland to determine whether thyroid function is normal. If the thyroid has absorbed all of the
iodine, what percentage of the original amount should be detected?
(b) A patient returns to the medical clinic 25 hours after having received an injection of 133I. The medical technician scans the patient’s thyroid gland and
detects the presence of 41.3% of the original iodine. How much of the original 133I remains in the rest of the patient’s body?
SOUND LEVELS
42. A decibel, named for Alexander Graham Bell, is the smallest increase of the loudness of sound that is detectable by the human ear. In physics, it is shown that
when two sounds of intensity I1 and I2 (watts/cm3) occur, the difference in loudness is D decibels, where
D 10 log10
II 1
2
When sound is rated in relation to the threshold of human hearing (I0 1012),
the level of normal conversation is about 60 decibels, while a rock concert may
be 50 times as loud (110 decibels).
328
Chapter 4
Exponential and Logarithmic Functions
(a) How much more intense is the rock concert than normal conversation?
(b) The threshold of pain is reached at a sound level roughly 10 times as loud as
a rock concert. What is the decibel level of the threshold of pain?
SPY STORY
43. Having defused the bomb in Problem 20 of Section 3.5, the spy returns home,
only to learn that his best friend, Sigmund (“Siggy”) Leiter, has been murdered.
The police say that Siggy’s body was discovered at 1 P.M. on Thursday, stuffed
in a freezer where the temperature was 10°F. He is also told that the temperature
of the corpse at the time of discovery was 40°F, and he remembers that t hours
after death a body has temperature
T Ta (98.6 Ta)(0.97)t
where Ta is the air temperature adjacent to the body. The spy knows the dark deed
was done by either Ernst Stavro Blohardt or André Scélérat. If Blohardt was in
jail until noon on Wednesday and Scélérat was seen in Las Vegas from noon
Wednesday until Friday, who “iced” Siggy, and when?
LEARNING CURVE
44. In an experiment designed to test short-term memory,* L. R. Peterson and M. J.
Peterson found that the probability p(t) of a subject recalling a pattern of numbers and letters t seconds after being given the pattern is
p(t) 0.89[0.01 0.99(0.85)t]
(a) What is the probability that the subject can recall the pattern immediately
(t 0)?
(b) How much time passes before p(t) drops to 0.5?
(c) Sketch the graph of p(t).
MAGNITUDE OF EARTHQUAKES
AIR PRESSURE
45. On the Richter scale, the magnitude of an earthquake or intensity I is given by
ln I
.
R
ln 10
(a) Find the intensity of the 1906 San Francisco earthquake, which measured 8.3
on the Richter scale.
(b) How much more intense was the San Francisco earthquake of 1906 than the
devastating 1995 earthquake in Kobe, Japan, which measured 7.1?
46. The air pressure f(s) at a height of s meters above sea level is given by
f(s) e0.000125s atmospheres
(a) The atmospheric pressure outside an airplane is 0.25 atmosphere. How high
is the plane?
(b) A mountain climber decides she will wear an oxygen mask once she has reached
an altitude of 7,000 meters. What is the atmospheric pressure at this altitude?
* L. R. Peterson and M. J. Peterson, “Short-Term Retention of Individual Verbal Items,” Journal of
Experimental Psychology, Vol. 58 (1959), pages 193–198.
Chapter 4 ■ Section 2
Logarithmic Functions
329
POPULATION GROWTH
47. Based on the estimate that there are 10 billion acres of arable land on the earth
and that each acre can produce enough food to feed 4 people, some demographers
believe that the earth can support a population of no more than 40 billion people.
The population of the earth was approximately 3 billion in 1960 and 4 billion in
1975. If the population of the earth were growing exponentially, when would it
reach the theoretical limit of 40 billion?
POPULATION GROWTH
48. According to a logistic model based on the assumption that the earth can support
no more than 40 billion people, the world’s population (in billions) t years after
40
1960 is given by a function of the form P(t) , where C and k are pos1 Cekt
itive constants. Find the function of this form that is consistent with the fact that
the world’s population was approximately 3 billion in 1960 and 4 billion in 1975.
What does your model predict for the population in the year 2000? Check the
accuracy of the model by consulting an almanac.
ECOLOGY
49. According to the Bouger-Lambert law, a beam of light that strikes the surface of
a body of water with intensity I0 will have intensity I I0ekx at a depth of x
meters. The constant k, called the absorption coefficient, depends on such things
as the purity of the water and the wavelength of the beam of light.*
(a) Suppose a particular beam of light is only 5% as intense at a depth of 3 meters
as it is at the surface. Use this information to find k and then determine the
depth at which the intensity is 1% of the intensity at the surface.
(b) Write a paragraph on how the Bouger-Lambert law explains why plant life
typically exists only in the top 10 meters of a lake or sea. You may wish to
begin your research with the reference cited in this problem.
ARCHAEOLOGY
50. “Lucy,” the famous prehuman whose skeleton was discovered in Africa, has been
found to be approximately 3.8 million years old.
(a) Approximately what ratio of 14C to 12C would you expect to find if you tried
to apply carbon dating to Lucy? Why would this be a problem if you were
actually trying to “date” Lucy?
(b) In practice, carbon dating works well only for relatively “recent” samples—
those that are no more than approximately 50,000 years old. For older samples, such as Lucy, variations on carbon dating have been developed, such as
potassium-argon and rubidium-strontium dating. Read an article on alternative
dating methods and write a paragraph on how they are used.†
* E. Batschelet, Introduction to Mathematics for Life Scientists, 2nd ed., Springer-Verlag, New York,
1976, page 318.
† A good place to start your research is the article by Paul J. Campbell, “How Old Is the Earth,” UMAP
Modules 1992: Tools for Teaching, Consortium for Mathematics and Its Applications, Arlington, MA,
1993.
330
RADIOACTIVE DECAY
Exponential and Logarithmic Functions
51. A radioactive substance decays exponentially with half-life . Suppose the amount
of the substance initially present (when t 0) is Q0.
(a) Show that the amount of the substance that remains after t years will be
Q(t) Q0e(ln 2/)t.
(b) Find a number k so that the amount in part (a) can be expressed as Q(t) Q0(0.5)kt.
52. In each case, use one of the laws of exponents to prove the indicated law of
logarithms.
u
(a) The quotient rule: ln ln u ln v
v
(b) The power rule: ln ur r ln u
y
y=x
(b, a)
53. Let a and b be any positive numbers other than 1. Show that
(a, b)
0
Chapter 4
(loga b)(logb a) 1
x
[Hint: Use the conversion formula.]
54. Show that the reflection of the point (a, b) in the line y x is (b, a). [Hint: Show
that the line joining (a, b) and (b, a) is perpendicular to y x and that the distance from (a, b) to y x is the same as the distance from y x to (b, a).]
PROBLEM 54
55. Use the graphing utility to graph y 10x, y log10 x, and y x on the same
coordinate axes (use [5, 5]1 by [5, 5]1). How are these graphs related?
In Problems 56 through 59 solve for x.
56. x ln (3.42 108.1)
57. 3,500e0.31x e3.5x
1 257e1.1x
58. e0.113x 4.72 7.031 x
59. ln (x 3) ln x 5 ln (x2 4)
SEISMOLOGY
60. The magnitude formula for the Richter scale is
M
2
E
log10
3
E0
where E is the energy released by the earthquake (in joules), and E0 104.4 joules
is the energy released by a small reference earthquake used as a standard of measurement.
(a) The 1906 San Francisco earthquake released approximately 5.96 1016 joules
of energy. What was its magnitude on the Richter scale?
(b) How much energy was released by the Indian earthquake of 1993, which measured 6.4 on the Richter scale?
Chapter 4 ■ Section 3
RADIOLOGY
3
Differentiation
of Logarithmic
and
Exponential
Functions
Differentiation of Logarithmic and Exponential Functions
331
61. The radioactive isotope gallium-67 (67Ga), used in the diagnosis of malignant
tumors, has a half-life of 46.5 hours. If we start with 100 milligrams of the isotope,
how many milligrams will be left after 24 hours? When will there be only 25 milligrams left? Answer these questions by first using the graphing utility to graph
an appropriate exponential function and then using the trace and zoom features.
Exponential functions and logarithmic functions both have simple derivatives. In this
section, we will obtain derivative formulas for these functions and use them to analyze a few practical problems. We begin with the derivative formula for ln x.
The Derivative of ln x
■
d
1
(ln x) for x 0
dx
x
The derivation of this formula is given at the end of this section, after we examine a variety of examples involving its use.
EXAMPLE 3.1
Graph y ln x using a modified
decimal window, [0.7, 8.7]1
by [3.1, 3.1]1. Choose a value
of x and construct the tangent
line to the curve at this x. Observe how close the slope of the
1
tangent line is to . Repeat this
x
for several additional values
of x.
Differentiate the function f(x) x ln x.
Solution
Combine the product rule with the formula for the derivative of ln x to get
f(x) x
x ln x 1 ln x
1
EXAMPLE 3.2
3
Differentiate f(x) ln x2
.
x4
Solution
3
First, since x2 x2/3, the power rule for logarithms allows us to write
3
f(x) 2/3
ln x
ln x
x4
x4
2
2
ln x
3
4
x