```Chapter 3
A1
Glencoe Precalculus
DATE
Before you begin Chapter 3
Exponential and Logarithmic Functions
Anticipation Guide
PERIOD
A
D
D
2. Logarithmic functions and exponential functions are inverses
of each other.
3. A logarithm with base 10 or log 10 is called a natural
logarithm.
4. A logarithm with base e or log e is called a common logarithm
and is denoted by ln.
D
8. Exponential models apply to any situation where the change
is proportional to the initial size of the quantity being
considered.
After you complete Chapter 3
A
7. A logistic function is shown on a graph that changes rapidly
and then has a horizontal asymptote.
6. Logarithmic functions do not have the one-to-one property.
A
D
A
1. For most real-world applications involving exponential
functions, the most commonly used base is not 2 or 10 but an
irrational number e.
5. Exponential functions have the one-to-one property.
STEP 2
A or D
Statement
Chapter 3
3
Chapter Resources
3/18/09 5:18:44 PM
Glencoe Precalculus
• For those statements that you mark with a D, use a piece of paper to write an example
of why you disagree.
• Did any of your opinions about the statements change from the first column?
• Reread each statement and complete the last column by entering an A or a D.
Step 2
STEP 1
A, D, or NS
• Write A or D in the first column OR if you are not sure whether you agree or disagree,
write NS (Not Sure).
• Decide whether you Agree (A) or Disagree (D) with the statement.
Step 1
3
NAME
0ii_004_PCCRMC03_893804.indd Sec1:3
DATE
PERIOD
27
-3
9
-2
3
-1
1
0
1
−
9
1
−
3
27
1
−
3
Asymptote: x-axis
Range: (0, ∞)
2
1
x→∞
x
0
y
(
x
2
)
5
decreasing: (-∞, ∞)
increasing: (-∞, ∞)
005_032_PCCRMC03_893804.indd 5
Chapter 3
x → -∞
lim
f(x) = 0;
x→∞
x → -∞
lim
f(x) = ∞;
x→∞
x
asymptotes: y = 1;
end behavior:
f(x) = 1 and
lim
0
y
2. k(x) = e-2x
D = (-∞, ∞);
R = (0, ∞);
intercept: (0, 1);
asymptote: x-axis;
end behavior:
lim
f(x) = ∞ and
D = (-∞, ∞);
R = (1, ∞);
1
intercept: 0, 1 −
;
0
y
1. h(x) = 2x - 1 + 1
x
Lesson 3-1
10/1/09 1:42:35 PM
Glencoe Precalculus
Sketch and analyze the graph of each function. Describe its domain,
range, intercepts, asymptotes, end behavior, and where the function
is increasing or decreasing.
Exercises
Decreasing: (-∞, ∞)
x → -∞
End behavior: lim f(x) = ∞ and lim f(x) = 0
Intercept: (0, 1)
Domain: (-∞, ∞)
f(x)
x
()
1
. Describe
Sketch and analyze the graph of f(x) = −
3
its domain, range, intercepts, asymptotes, end behavior, and where
the function is increasing or decreasing.
Example
An exponential function with base b has the
form f(x) = abx, where x is any real number and a and b are real number
constants such that a ≠ 0, b is positive, and b ≠ 1. If b > 1, then the function
is exponential growth. If 0 < b < 1, then the function is exponential decay.
Exponential Functions
Study Guide and Intervention
Exponential Functions
3-1
NAME
Answers (Anticipation Guide and Lesson 3-1)
Exponential Functions
Study Guide and Intervention
DATE
(continued)
PERIOD
N = N0ekt
N is the final amount, N0 is
the initial amount, r is the
rate of growth or decay, r is
time, and e is a constant.
N = N0(1 + r)t
N is the final amount, N0 is
the initial amount, r is the
rate of growth or decay, and
t is time.
nt
P is the principal or initial
investment, A is the final
amount of the investment,
r is the annual interest rate,
n is the number of times
interest is compounded each
year, and t is the number
of years.
r
A = P1 + −
n
Compound Interest
Use a calculator.
N0 = 100, r = 0.25, t = 6
Exponential Growth Formula
A2
Glencoe Precalculus
005_032_PCCRMC03_893804.indd 6
Chapter 3
6
Glencoe Precalculus
3. BIOLOGY The number of rabbits in a field showed an increase of 10% each month over
the last year. If there were 10 rabbits at this time last year, how many rabbits are in the
field now? 33 rabbits
2. ENERGY In 2007, it is estimated that the United States used about 101,000 quadrillion
thermal units. If U.S. energy consumption increases at a rate of about 0.5% annually,
what amount of energy will the United States use in 2020? about 107,766
with continuous
1. FINANCIAL LITERACY Compare the balance after 10 years of a \$5000 investment
earning 8.5% interest compounded continuously to the same investment compounded
quarterly. continuously: \$11,698.23, quarterly \$11,594.52, \$103.71 more
Exercises
Example 2
FINANCIAL LITERACY Lance has a bank account that will allow him
to invest \$1000 at a 5% interest rate compounded continuously. If there are no
other deposits or withdrawals, what will Lance’s account balance be after 10 years?
A = Pert
Continuous Compound Interest Formula
= 1000e(0.05)(10)
P = 1000, r = 0.05, and t = 10
≈ 1648.72
Simplify.
With continuous compounding, Lance’s account balance after 10 years will be \$1648.72.
There will be about 381 cells in the colony in 6 weeks.
N = N0(1 + r)t
= 100(1 + 0.25)6
≈ 381.4697266
Example 1
BIOLOGY A researcher estimates that the initial population of a
colony of cells is 100. If the cells reproduce at a rate of 25% per week, what is the
expected population of the colony in six weeks?
Continuous Exponential
Growth or Decay
Exponential Growth or
Decay
10/1/09 1:43:46 PM
Chapter 3
Exponential Growth and Decay Many real-world situations can be
modeled by exponential functions. One of the equations below may apply.
3-1
NAME
Exponential Functions
Practice
DATE
PERIOD
4
−4 0
4
8x
decreasing: (-∞, ∞)
and
005_032_PCCRMC03_893804.indd 7
Chapter 3
7
4. FINANCE Determine the amount of money in a savings account that
provides an annual rate of 4% compounded monthly if the initial deposit
is \$1000 and the money is left in the account for 5 years. \$1221.00
b. Predict the population in 2020 and 2030. Assume a steady rate
of increase. 335,319,934 in 2020 and 366,024,459 in 2030
Lesson 3-1
3/18/09 6:00:31 PM
Glencoe Precalculus
a. Let t be the number of years since 2000. Write a function that models
the annual growth in population in the U.S. N = 281421906(1 + 0.0088)t
3. DEMOGRAPHICS In 2000, the number of people in the United States
was 281,421,906. The U.S. population is estimated to be growing at
0.88% annually.
increasing: (-∞, ∞)
and
f(x) = -2
lim
x → -∞
lim f(x) = -∞;
x→∞
asymptote: y = -2;
asymptote: x-axis;
end behavior:
x-intercept: none;
x-intercept: none;
f(x) = 0
lim
x → -∞
lim f(x) = ∞;
x→∞
y-intercept: (0, -2.2);
end behavior:
R = (-∞, -2);
R = (0, ∞);
1
;
y-intercept: 0, −
( 2)
y
5
D = (-∞, ∞);
−8
−8
−8
8x
−4
4
4
8
1 x
2. h(x) = - −
e -2
−4
−4 0
y
D = (-∞, ∞);
−8
8
1. f(x) = 2 x - 1
Sketch and analyze the graph of each function. Describe its domain,
range, intercepts, asymptotes, end behavior, and where the function
is increasing or decreasing.
3-1
NAME
Chapter 3
Exponential Functions
A3
8
PERIOD
215
210
220
2011
225
2021
230
2031
Glencoe Precalculus
3/18/09 6:00:37 PM
Glencoe Precalculus
No; week 21 has 220 seconds of
homework or about 291 hours of
homework.
5. If your precalculus teacher offers to give
you 1 second of homework for the first
week of school and double the amount of
homework each week until the end of the
school year (i.e. 2 seconds the second
week), should you say yes? Explain.
2001
1991
−t
P = 1 22
c. A 1971 computer could manage one
process per second. Every two years,
the number of processes also doubles
on a computer. Write an equation to
calculate the number of processes a
computer can manage every second
in each year after 1971. Then
complete the table below.
1.1 × 109
b. Approximately how many transistors
were on one circuit in 2009?
N = 2100 2 2
−t
a. If there were about 2100 transistors
on every circuit in 1971, write an
exponential equation to model the
number of transistors in a given year
t after 1971.
005_032_PCCRMC03_893804.indd 8
Chapter 3
the higher monthly rate. To check,
the balance is \$1120.45 versus
\$1118.51; after 5 years, the
balance is \$1328.87 versus
\$1323.13.
3. FINANCAL LITERACY You have \$1000
to put into the bank. One bank offers a
5.7% interest rate compounded monthly.
Another bank offers 5.6% compounded
continuously. Which would you choose to
make the most money after 2 years? after
5 years? Explain.
P(5) = 388; P(10) = 1503;
P(24) = 66,781;
P(72) = 2.98 × 1010
c. Find the number of bacteria in
5, 10, 24, and 72 hours. Round to
the nearest whole number.
100
b. What was the initial number
of bacteria?
27.1% per hour
a. Determine the growth rate.
2. BIOLOGY Suppose a certain type of
bacteria reproduces according to the
model P(t) = 100e0.271t, where t is time in
hours and P(t) is the number of bacteria.
\$1578.18
DATE
4. TECHNOLOGY In 1965, Gordon Moore
stated that since the invention of the
integrated circuit in 1958, the number of
transistors that can be placed on that
circuit has doubled every two years. This
statement has been true to the present
day. Almost all measures of computing
power come from this statement so that
we can say that the computing power
doubles nearly every two years.
Word Problem Practice
1. FINANCIAL LITERACY Suppose Jamal
has a savings account with a balance of
\$1400 at a 4% interest rate compounded
monthly. If there are no other deposits or
withdrawals, what will be Jamal’s
account balance in three years?
3-1
NAME
Enrichment
DATE
PERIOD
2
3
3!
3
3!
2
2
2!
4
4
5
5!
3
3!
2
4
5
5!
n
n!
2
3
3!
3
3!
2
2!
4
4
4!
5
4!
5
5!
6
6!
005_032_PCCRMC03_893804.indd 9
Chapter 3
1. e1.5 4.48
2. e2 7.38
9
3. e2.1 8.16
Lesson 3-1
10/1/09 10:40:06 AM
Glencoe Precalculus
4. e2.2 9.02
Use the Maclaurin polynomials to approximate each of the following
powers of e to the hundredths place.
Exercises
3
3!
2
2!
4
5!
1
1
1
1
1
+−
+−
+−
+−
≈ 2.7181
e1 ≈ 1 + 1 + −
4!
1
1
1
1
+−
+−
+−
≈ 2.7166
e1 ≈ 1 + 1 + −
3
3!
2
2!
1
1
1
+−
+−
≈ 2.7083
e1 ≈ 1 + 1 + −
2!
1
1
+−
≈ 2.6667
e1 ≈ 1 + 1 + −
2
2!
1
= 2.5
e1 ≈ 1 + 1 + −
e1 ≈ 1 + 1 = 2
When x = 1, these polynomials can be used to approximate e1 or e to the
thousandths place. Remember e ≈ 2.718.
4!
x
x
x
x
x
+−
+−
+−
+…+−
.
ex ≈ 1 + x + −
2!
By adding more and more terms, you get the general Maclaurin polynomial
4!
x
x
x
x
+−
+−
+−
ex ≈ 1 + x + −
4!
x
x
x
+−
+−
ex ≈ 1 + x + −
2!
3
3!
2
2!
x
x
+−
ex ≈ 1 + x + −
2!
x
ex ≈ 1 + x + −
ex ≈ 1 + x
It can be shown using calculus that the function e x can be approximated by
what are known as Maclaurin polynomials as follows.
Maclaurin Polynomials
3-1
NAME
Logarithmic Functions
Study Guide and Intervention
DATE
exponent
-2
A4
25
Equality Prop. of Exponents
1
−
= 5-2
25
Write in exponential form.
1
Let log5 −
= y.
Evaluate each logarithm.
ln ex = x
-4
eln x = x
1
2
2
x
2
6. e ln x
2
3. 3
2
log3 2
10log 13 = 13
c. 10log 13
.
because 3 = √3
−
10log x = x
Equality Prop. of Exponents
1
−
3 2 = √
3
Write in exponential form.
Let log3 √
3 = y.
1
=−
Therefore, log3 √3
2
exponent
Glencoe Precalculus
005_032_PCCRMC03_893804.indd 10
Chapter 3
10
Glencoe Precalculus
7. FINANCIAL LITERACY Ms. Dasilva has \$3000 to invest. She would like to invest in an
account that compounds continuously at 6%. Use the formula ln A - ln P = rt, where A
is the current balance, P is the original principal, r is the rate as a decimal, and t is the
time in years. How long will it take for her balance to be \$6000? 11.55 years
2
1
5. log3 −
81
5x
1
4. log6 36
2. 10log 5x
1. log7 7
eln 5 = 5
b. eln 5
1
−
3y = 3 2
1
y=−
base
blogbx = x, x > 0
log3 √
3=y
3y = √
3
b. log3 √
3
Evaluate each expression.
Evaluate each logarithm.
Exercises
ln e7 = 7
a. ln e7
Example 2
because 5-2
25
1
=−
.
25
1
Therefore, log5 −
= -2
5 =5
y = -2
y
1
a. log5 −
25
1
log5 −
=y
25
1
y
5 =−
25
Example 1
The following properties are also useful.
logb 1 = 0
logb b = 1
logb bx = x
base
by = x
logb x = y
if and only if
Exponential Form
Logarithmic Form
If b > 0, b ≠ 1, and x > 0, then
The inverse relationship
between logarithmic functions and exponential functions can be used to
evaluate logarithmic expressions.
PERIOD
10/1/09 10:45:06 AM
Chapter 3
Logarithmic Functions and Expressions
3-2
NAME
DATE
(continued)
PERIOD
-2
0.028
-1
0.17
0
1
1
6
2
36
4
x→∞
0
4
8
12 x
005_032_PCCRMC03_893804.indd 11
Chapter 3
−2.0
−1.0
0
1.0
2.0
y
2
4
1. g(x) = log3 x
6
x
11
Lesson 3-2
10/1/09 1:46:02 PM
Glencoe Precalculus
= -∞;
decreasing (2, ∞)
x → 2+
increasing (0, ∞)
7x
lim
5
x → ∞ f(x)
3
D = (2, ∞);
R = (-∞, ∞);
y-intercept: none;
x-intercept: (3, 0);
asymptote: x = 2;
end behavior:
lim f(x) = ∞ and
lim
x → ∞ f(x) = ∞;
x→0
−1.0
0 1
1.0
2.0
y
2. h(x) = -log3 (x - 2)
D = (0, ∞);
R = (-∞, ∞);
y-intercept: none;
x-intercept: (1, 0);
asymptote: y-axis;
end behavior:
lim f(x) = -∞ and
+
Sketch and analyze the graph of each function below. Describe its
domain, range, intercepts, asymptotes, end behavior, and where the
function is increasing or decreasing.
Exercises
Increasing: (0, ∞)
x → 0+
End behavior: lim f(x) = -∞ and lim f(x) = ∞
Asymptote: y-axis
Since the functions are inverses, you can obtain the graph of f(x) by plotting
the points (f -1(x), x).
y
Domain: (0, ∞)
12
Range: (-∞, ∞)
8
x-intercept: (1, 0)
f -1(x)
x
Create a table of values for the inverse of the function, the exponential
function f -1(x) = 6x.
Example
Sketch and analyze the graph of f(x) = log6 x.
Describe its domain, range, intercepts, asymptotes, end behavior,
and where the function is increasing or decreasing.
The inverse of f(x) = bx is called
the logarithmic function with base b, or f(x) = logb x, and read f of x equals
the log base b of x.
Logarithmic Functions
Study Guide and Intervention
Graphs of Logarithmic Functions
3-2
NAME
Chapter 3
DATE
2
9. log12 144
3
5
−
6. log8 32
4
3. log8 4096
PERIOD
A5
0
y
P0
12
Glencoe Precalculus
10/23/09 6:40:38 PM
Glencoe Precalculus
x
005_032_PCCRMC03_893804.indd 12
Chapter 3
investment. An investment of \$4000 was made in 2005 and had a value of \$7500
in 2010. What was the average growth rate of the investment? 12.6%
t
1
P
ln −
, where t is time in years, P is the present value, and P0 is the original
r=−
g(x)
f(x)
1
.
down and compressed horizontally by a factor of −
2
13. INVESTMENTS The annual growth rate r for an investment can be found using
The graph of g(x) is the graph of f(x) shifted 2 units
(2)
x
f(x) = ln x, g(x) = ln −
-2
12. Use the graph of f to describe the transformation
that results in the graph of g. Then sketch the
graphs of f and g.
decreasing: (-∞, ∞)
decreasing: (-∞, ∞)
x→∞
R = (0, ∞);
y-intercept: (0, e) or (0, 2.72);
x-intercept: none;
asymptote: x-axis;
lim f(x) = ∞
end behavior: x →
-∞
lim
and
f(x) = 0;
8x
D = (-∞, ∞);
4
R = (0, ∞);
y-intercept: (0, 16);
x-intercept: none;
asymptote: x-axis;
lim f(x) = ∞
end behavior: x →
-∞
lim
and x → ∞ f(x) = 0;
8x
−4
−4 0
y
D = (-∞, ∞);
4
−8
4
11. g(x) = e
−8
−4 0
y
2x + 1
−6
−8
6
12
18
10. g(x) = 4
-x + 2
Sketch and analyze the graph of each function. Describe its domain,
range, intercepts, asymptotes, end behavior, and where the function is
increasing or decreasing.
0.014x
8. eln 0.014x
7. log6 216
3
18
5. 9
log 9 18
-3
10
4. 2 ln e5
3
1. log7 73
2. log10 0.001
Logarithmic Functions
Practice
Evaluate each expression.
3-2
NAME
P0
005_032_PCCRMC03_893804.indd 13
Chapter 3
not hit zero because there is an
asymptote at y = 0.
c. Will the population of the city ever
reach zero if the trend continues?
How do you know? Sample
b. What is the population in 2010
and 2025? 50,000 and 49,729
a. Is the population growing or
shrinking? shrinking
3. POPULATION The population P
of a certain city in any year t
since 2010 is given by the formula
P = -100 ln t + 50,000.
growth rate, t is time in years, P is the
present value, and P0 is the original
investment. If the original investment
was \$2500 in 2002 at a rate of 6.5%,
in what year could you withdraw
\$10,000? 2023
t
1
P
ln −
, where r is the annual
r=−
2. INVESTING The annual growth rate for
an investment can be found using
c. How much more energy is released in
an earthquake of magnitude 8.0 than
of magnitude 3.0?
1000 kilowatt hours
b. If an earthquake has magnitude 3.0,
the amount of energy released.
a. If the amount of energy released is
1 million kilowatt hours, what is the
13
Logarithmic Functions
DATE
PERIOD
5
10
15
30
k
Lesson 3-2
10/1/09 1:47:17 PM
Glencoe Precalculus
No; they need an interest rate of
13.9%.
that earns 4.5% interest compounded
continuously, will the investment
double in 5 years? If not, what
interest rate do they need?
parents invested \$5000 in an account
rate written as a decimal. If Tara’s
5. FINANCIAL LITERACY The doubling
time t, or the amount of time it takes for
an investment to double, is given
ln 2
by t = −
, where k is the interest
change is initially high, but
then it slows down.
d. What can you conclude about
the rate of change in logarithmic
functions?
c. On which day is the population
four times as high? 31
b. If the initial population is
25 bacteria, on which day will the
population triple? 10
p
3
1
25 40 49 60 75 84 99
t
2
a. Complete the table below. Round
to the nearest whole number.
4. BIOLOGY A certain strain of
bacteria has a population that can
be approximated by the formula
P = 50 log 3.2t, where P is the population
and t is the time elapsed
in days.
Word Problem Practice
1. SEISMOLOGY The Richter scale
measures the magnitude M of an
earthquake M = 0.67 log (0.37 E) + 1.46,
where E is the energy released by the
earthquake in kilowatt hours.
3-2
NAME
Enrichment
PERIOD
9
7
10%
4
20%
3
30%
9
36 18 12
D2(r)
7
10%
4
20%
2
30%
A6
Glencoe Precalculus
005_032_PCCRMC03_893804.indd 14
Chapter 3
14
Glencoe Precalculus
5. Suppose your grandparents had placed \$8000 dollars in an account when
you were born. If your account earns r% compounded annually, what
number would you divide by r to determine the approximate age at which
the account will be worth about \$1,000,000? If your account is earning 2%,
would it be reasonable to assume that you would ever be a millionaire in
13 years; 5 years; 2 years; The estimates are very close to
the calculated values.
4. Bankers have long used a rule known as the Rule of 72 to obtain a quick
estimation of the number of years it takes to double an investment
when compounded annually. They simply divide 72 by the interest rate to
obtain a fairly accurate answer. How many years are needed to double
an investment if it is earning 5.5%? 15%? 40%? How do the estimates
using the Rule of 72 compare to the times calculated using the formula?
The answers found in the table in Exercise 1 are very similar
to the answers found in the table in Exercise 2.
3. Compare the results from Exercises 1 and 2. What do you observe?
8%
6%
4%
2%
r
72
2. Now complete the table below by evaluating the function D2(r) = −
r.
8%
6%
D1(r)
4%
2%
35 18 12
r
1. Complete the table below. Find the number of years needed to double your
money at each of the given interest rates. Round your answers to the
nearest year.
ln (1 + r)
ln 2
, where r is written as a decimal.
given by D1(r) = −
of money invested and compounded annually at an interest rate of r% is
The function D1 that gives the number of years needed to double the amount
DATE
10/1/09 1:48:30 PM
Chapter 3
3-2
NAME
DATE
PERIOD
3y
8x
Expand ln −
.
2
5
3
005_032_PCCRMC03_893804.indd 15
Chapter 3
(x − 3)11
log9 −
32x5
4. 11 log9 (x − 3) − 5 log9 2x
Condense each expression.
2
log3 5 + 5 log3 r − −
log3 t
5r
2. log3 −
3
√t2
5
Expand each expression.
1
1. Evaluate 2 log3 27 + 4 log3 −
.
3
2
= ln 8 + 5 ln x − ln 3 − 2 ln y
15
4
5
5
Lesson 3-3
3/18/09 6:01:11 PM
Glencoe Precalculus
ln ( √
(2h − k)3 √
(2h + k)3 )
4
3
3
5. −
ln (2h − k) + −
ln (2h + k)
log (a - 2) + 6 log (b + 4) –
log 9 – 5 log (b - 2)
(a − 2)(b + 4)6
9(b − 2)
3. log −
5
Power Property
Product Property
Quotient Property
Simplify.
=4
3y
logx x = 1
= 3(3)(1) + 5(-1)(1)
= ln 8 + ln x5 − ln 3 − ln y2
Exercises
2
1
8 = 2 3; 2 -1 = −
Power Property
2
= 3(3 log2 2) + 5(-log2 2)
8x
= ln 8x5 − ln 3y2
ln −
2
5
Example 2
2
1
Evaluate 3 log2 8 + 5 log2 −
.
Power Property
Quotient Property
Product Property
1
= 3 log 2 23 + 5 log 2 2−1
3 log2 8 + 5 log 2 −
Example 1
• logb −xy = logb x − logb y
• logb xp = p logb x
• logb xy = logb x + logb y
If b, x, and y are positive real numbers, b ≠ 1, and p is a real number, then
the following statements are true.
Since logarithms and exponents have an
inverse relationship, they have certain properties that can be used to make
them easier to simplify and solve.
Properties of Logarithms
Study Guide and Intervention
Properties of Logarithms
3-3
NAME
Answers (Lesson 3-2 and Lesson 3-3)
Chapter 3
DATE
Properties of Logarithms
Study Guide and Intervention
(continued)
PERIOD
A7
Use a calculator.
Change of Base Formula
Evaluate each logarithm.
8. log 11 47 1.606
7. log 6 832 3.753
16
Use a calculator.
Change of Base Formula
Glencoe Precalculus
10/1/09 10:48:42 AM
Glencoe Precalculus
12. log0.5 420 -8.714
9. log 3 9 2
6. log 9 712 2.99
3. log 7 1094 3.60
≈ -2.10
3
1
log −
005_032_PCCRMC03_893804.indd 16
Chapter 3
11. log 12 4302 3.367
5. log 5 256 3.45
4. log 6 94 2.54
10. log8 256 2.667
2. log 3 17 2.58
3
log 10
log −1 10 = −
3
b. log −1 10
ln b
ln x
log b x = −
1. log 32 631 1.86
Evaluate each logarithm.
Exercises
≈ 2.81
ln 7
log 2 7 = −
ln 2
a. log2 7
Example
log b x = −
log x
log b
Many non-graphing calculators cannot be used for logarithms that are not
base e or base 10. Therefore, you will often use this formula, especially for
scientific applications. Either of the following forms will provide the correct
log a x
For any positive real numbers a, b, and x, a ≠ 1, b ≠ 1, log b x = −
.
log a b
If the logarithm is in a base that needs to be
changed to a different base, the Change of Base Formula is required.
Change of Base Formula
3-3
NAME
Properties of Logarithms
Practice
DATE
4
3
2
(x + 1) 3
√x
+5
10. log 2 −
3
3
3 log 8 (4x + 2) + log 8 (x − 4)
8. log8 [(4x + 2) 3(x − 4)]
2
1
2 ln x - −
ln (3x + 2)
√
3x + 2
x
6. ln −
3x − 5y
√
3
2
5
I0
1.774
21. log 6 24
1.151
18. log 100 200
2
log 2 −
(x + 4)
√x
log 3 −
xyz 3
2
3
−2.078
22. log −1 9.8
-0.301
19. log 0.01 4
1
16. log 3 y + log 3 x – −
log 3 x + 3 log 3 z
8x
log 3 −
2
√
x−4
14. log 3 8 + log 3 x – 2 log 3 (x + 4)
(5x + 6) 3
1
12. 3 log 2 (5x + 6) − −
log2 (x − 4)
005_032_PCCRMC03_893804.indd 17
Chapter 3
17
b. Macquarie Island region, in 2008, R = 7.1 about 12,589,254
a. Guam region, in 2008, R = 6.7 about 5,011,872
I 0 = 1. Find the intensity per unit of area for the following earthquakes.
Lesson 3-3
10/1/09 1:51:14 PM
Glencoe Precalculus
23. SEISMOLOGY The intensity of a shock wave from an earthquake is given by the
I
, where R is the magnitude, I is a measure of wave energy, and
formula R = log 10 −
-4.046
20. log 0.24 322
2.322
1
17. log −1 −
Evaluate each logarithm.
√x
log −
3
3yz 2
1
15. log y + log 3 − −
log(x) + 2 log z
6x
49
log 7 −
2
13. 2 - log 7 6 – 2 log 7 x
4x + y
ln −
2
1
11. −
ln (3x – 5y) – ln (4x + y)
Condense each expression.
1
1
log 13 3 + 4 log 13 x − −
log 13 (7x − 3) 3 log 2 (x + 1) − −
log 2 (x + 5)
9. log 13
3x 4
−
3
√
7x − 3
3 + 3 log 2 x + log 2 (x + 1)
7. log2 [(2x) 3(x + 1)]
1
ln (x + 1) − −
ln (x − 5)
5. ln −
4
x+1
√
x−5
Expand each expression.
10000
729
4. ln −
6 ln 3 − 4 ln 10
ln 10 − 2 ln 3
2. ln 27000 3 ln 3 + 3 ln 10
10
3. ln −
9
PERIOD
1. ln 300 ln 3 + 2 ln 10
Express each logarithm in terms of ln 10 and ln 3.
3-3
NAME
A8
Glencoe Precalculus
005_032_PCCRMC03_893804.indd 18
Chapter 3
(
)
r
log E = n log 1 + −
n -1
4. The effective annual yield E for an
account compounded n times per year
at n percent is modeled by the equation
r n
log E = log (1 + −
n ) - 1. Rewrite the
equation using the properties of
logarithms.
6.23 characters, or 7 characters
will make 7.2 × 10 16 tries
necessary
3. TECHNOLOGY Computer hackers are
often very interested in passwords to
256 keys someone can use to make a
password character. In order to guess a
password of length L, a hacker would
have to try N different combinations to
access the system. If the formula is
L = log 256 N, how long do you have to
guarantee at least 10 15 tries by hackers?
2. BIOLOGY A formula to express the
amount of time it takes to double a
certain colony of bacteria is rt = ln 2,
where r is the growth rate and t is the
doubling time in hours. How long will it
take to double a bacteria colony if it
grows at a rate of 5% per hour? 13.9 h
18
Properties of Logarithms
(
PERIOD
P
)
Glencoe Precalculus
d. Since 2004, the likelihood of that
impact has dropped to 1 in 45,000.
What is the new Palermo Scale
number for 99942 Apophis? -2.0
c. The threshold for monitoring an
object with a potential to collide with
Earth is a Palermo Scale number
between −2 and 0. There is some
concern if an object has a number
higher than 0. In 2004, the object
99942 Apophis broke the record for
the highest Palermo Scale number. It
had a 2.7% chance of collision with
Earth in 2029 with an energy of
26,000 Megatons. How high was the
Palermo Scale number? 1.1
b. What is the Palermo Scale
number for an asteroid which has a
12% chance to strike the Earth in
20 years with an energy of
0.45 Megatons? -1.98
5
P = log P i − log 0.3 +
4
−
log E − log T
a. Expand the formula using the
properties of logarithms.
P is the Palermo Scale number, P i is the
probability of impact estimated by the
path of the object, E is the energy in
Megatons of TNT of the impact, and T
is the time in years away of the
estimated impact.
0.3E 5 T
−
i
occurrence is P = log −
, where
−4
5. ASTRONOMY The Palermo Technical
Impact Hazard Scale is a logarithmic
scale used by astronomers to rate the
potential danger from asteroids colliding
with the Earth and causing a large
amount of damage. The formula to
measure the probability of such an
Word Problem Practice
DATE
3/18/09 6:01:25 PM
Chapter 3
1. CHEMISTRY A useful way to express the
half-life of a radioactive object is in terms
of the time it will take to decay to a
percentage P of its original value given
by the formula t = −T log 2 P, where T is
the half-life and t is the time elapsed to
the percentage in years. How long will it
take Carbon-14 to decay to 20% of its
original amount if the half-life is
5730 years? 13,305 yr
3-3
NAME
Enrichment
DATE
PERIOD
⎪x ⎥
)
1 + √
1 - x2
x+1
x-1
sech-1 x = ln −
; (0, 1]
x
2
1
coth-1 x = −
ln − ; (-∞, -1) ∪ (1, ∞)
cosh-1 x = ln (x + √
x 2 - 1 ); [1, ∞)
ex + e-x
2
)
-2x
005_032_PCCRMC03_893804.indd 19
Chapter 3
19
Lesson 3-3
3/18/09 6:01:29 PM
Glencoe Precalculus
Show that the f(x) = cosh x and g(x) = cosh-1 x are inverses. See students’ work.
Exercise
2
-x 2
)
=x
-x
=x
2
= ln (ex)
x
2
2x
=−
(
x
e -e
= ln −
+−
x + √x
+1
−
2
2
2x (x + √
x2 + 1 )
=−
2
2
1
ln −
2x
-x 2
e -2+e
) + −44 )
(−
√
4
x
e -e
−
+ 1)
√(
2 )
(e + e )
√
ex - e-x
= ln −
+−
2
2
2
2
1
x + √
x2 + 1 - −
x + √
x2 + 1
= −−
2
(
(
(
ex - e-x
g(f(x)) = ln −
+
ex - e-x
= ln −
+
2
ln (x + √
x +1)
x +1 )
- e ( x+ √
= e−−
2
ln (x + √
x +1)
x +1)
- e -ln (x + √
f(g(x)) = e−−
2
tanh x
1
coth x = −
,x≠0
Show that f(x) = sinh x and g(x) = sinh-1 x are inverses.
Show that f(g(x)) = g(f(x)) = x.
Example
√
1 + x2
1
csch-1 x = ln −
x + − ; (-∞, 0) ∪ (0, ∞)
(
1
tanh-1 x = −
ln − ; (-1, 1)
1+x
1-x
x 2 + 1 ); (-∞, ∞)
sinh-1 x = ln (x + √
2
cosh x
sinh x
tanh x = −
Definitions of the Inverse Hyperbolic Functions
1
sech x = −
cosh x
1
,x≠0
csch x = −
sinh x
2
Definitions of the Hyperbolic Functions
x
+ e-x
cosh x = e−
2
-x
-e
sinh x = e−
x
Vincenzo Riccati introduced the hyperbolic functions in the mid-18th century.
They were first used to compare the area of a semicircular region with the
area of a region under a hyperbola. These functions have properties similar to
the trigonometric functions you will study in Chapters 4 and 5. The
definitions of the hyperbolic functions and their inverses are given below.
Hyperbolic Functions and Their Inverses
3-3
NAME
Chapter 3
DATE
Exponential and Logarithmic Equations
Study Guide and Intervention
PERIOD
A9
x ≈ 2.68
log 19
x=−
log 3
e2x - 3ex + 2 = 0
u2 - 3u + 2 = 0
(u - 2)(u - 1) = 0
u = 2 or u = 1
ex = 2 or ex = 1
x = ln 2 or 0
8
5
Glencoe Precalculus
10/1/09 1:52:47 PM
Glencoe Precalculus
005_032_PCCRMC03_893804.indd 20
20
8. e 19x = 23 0.17
7. 32x = 6x − 1 −4.42
Chapter 3
calculator.
Divide by 8 and use a
Subtract 1 from each side.
ln e = 1
Power Property
Take the ln of both sides.
4. 2e2x + 12ex - 54 = 0 1.10
(8)
1
= −
11 − x
6. 2.4ex − 6 = 9.3 7.35
2
1
−
(4)
1
2. −
2x − 1
5. 92x = 12 0.57
3. 43x - 2
1 2x
=−
2
1. 9x = 33x - 4 4
logarithm of each side.
Take the natural
Substitute for u.
Solve.
Factor.
Original equation
b. e8x + 1 - 6 = 1
e8x + 1 = 7
Take the log of both sides.
ln e8x + 1 = ln 7
Power Property
(8x + 1) ln e = ln 7
Divide each side by log 3.
8x + 1 = ln 7
Use a calculator. Check this
8x = ln 7 − 1
solution in the original equation.
ln 7 − 1
x=−
≈ 0.12
Solve each equation. Round to the nearest hundredth.
Exercises
Subtract x from each side.
One-to-One Property
Power of a Power
16 = 4
2
Original equation
b. Solve e2x − 3ex + 2 = 0.
Solve each equation. Round to the nearest hundredth if necessary.
log 3x = log 19
x log 3 = log 19
a. 3x = 19
Example 2
4x - 1 = 16x
4x - 1 = (42)x
4x - 1 = 42x
x - 1 = 2x
-1 = x
a. Solve 4x − 1 = 16x.
Example 1
can express both sides of the equation as an exponent with the same base.
Then use the property to set the exponents equal to each other and solve. If
the bases are not the same, you can exponentiate each side of an equation
and use logarithms to solve the equation.
One-to-One Property of
Exponential Functions: For b > 0 and b ≠ 1, bx = by if and only if x = y.
Solve Exponential Equations
3-4
NAME
DATE
PERIOD
(continued)
Exponential and Logarithmic Equations
Study Guide and Intervention
Factor.
Solve.
7
12
4. log3 3x = log3 36
005_032_PCCRMC03_893804.indd 21
Chapter 3
6. ln x + ln (x + 16) = ln 8 + ln (x + 6) 4
21
5. log (16x + 2) + log (20x - 2) = log (319x2 + 9x - 2) 2
no solution
3. log (x + 1) + log (x - 3) = log (6x2 - 6)
4
1. log 3x = log 12
yields logs of negative numbers.
Therefore, -2 is extraneous.
x = 8 log2 (8 − 6) = 5 − log2 [2(8)]
log2 2 = 5 − log2 16, which is true.
Therefore, x = 8.
x = −2 log2 (−2 − 6) = 5 − log2 [2(-2)]
CHECK
Lesson 3-4
3/18/09 6:01:38 PM
Glencoe Precalculus
2. log12 2 + log12 x = log12 (x + 7)
exponential form.
Expand.
Solve each logarithmic equation.
Exercises
2x2 − 12x − 32 = 0
2(x - 8)(x + 2) = 0
x = 8 or -2
Rewrite in
Product Property
Rearrange the logs.
Original equation
Solve log2(x − 6) = 5 - log2 2x.
Use a calculator.
Divide each side by 4.
Write in exponential form. (Use 5 as the base when exponentiating.)
log2 (x - 6) = 5 - log2 2x
log2 (x - 6) + log2 2x = 5
log2 (2x(x - 6)) = 5
2x(x - 6) = 2 5
Example 2
x = 3906.25
4
Divide each side by 2.
Original equation
Solve 2 log5 4x − 1 = 11.
2 log5 4x − 1 = 11
2 log5 4x = 12
log5 4x = 6
4x = 5 6
56
x=−
Example 1
can express both sides of the equation as a logarithm with the same base.
Then convert both sides to exponential form, set the exponents equal to each
other and solve.
One-to-One Property of Exponential
Functions For b > 0 and b ≠ 1, logbx = logby if and only if x = y.
Solve Logarithmic Equations
3-4
NAME
6
5
A10
2
16. log5 (x + 4) + log5 x = log5 12 2
no solution
14. log (2x + 1) + log (x − 4) = log (2x2 - 4)
20. 4 = 1056 1.67
23. 6 2x - 1 = 18 1.31
19. 2 = 1210 1.14
22. 3 x − 8 + 2 = 38 11.26
- 4 = 19 −1.05
Glencoe Precalculus
005_032_PCCRMC03_893804.indd 22
Chapter 3
22
26. FINANCIAL LITERACY If \$500 is deposited in a savings account
providing an annual interest rate of 5.6% compounded quarterly,
how long will it take for the account to be worth \$750? 7.29 years
Glencoe Precalculus
24. 2 3 + 2x = 130 2.01
21. 5
x+3
25. BANKING Ms. Cubbatz invested a sum of money in a certificate of
deposit that earns 8% interest compounded continuously. The formula
for calculating interest that is compounded continuously is A = Pe rt.
If Ms. Cubbatz made the investment on January 1, 2005, and the
account was worth \$12,000 on January 1, 2009, what was the original
amount in the account? \$8713.79
9x
3x
Solve each equation. Round to the nearest hundredth.
no solution
17. log (x + 1) + log (x − 3) = log (6x - 6) 18. ln 0.04x = -8 ≈0.008
15. 2 -4x + 1 = 3 2x - 3 ≈0.80
0 and ≈0.811
13. 4e 2x - 13e x + 9 = 0
≈0.28 and ≈-0.23
12. 6 ln (x + 2) - 3 = 21 ≈52.6
≈-1.639
11. 6e6x - 17e 3x + 7 = 0
=6
-2x - 3
10. 4
x+2
40
9. ln (x - 5) + ln 4 = ln x - ln 2 −
7
8. 3e4x - 9e2x - 15 = 0 ≈0.7167
17
2
4. e2x - ex - 6 = 0 ≈1.099
7. ln (2x - 1) = ln 16 −
= 27x - −
1
6. ln −
e = x -1
x+3
5. logx 64 = 3 4
1
3. −
(9)
1. 5x = 125 x-2 3
PERIOD
2. log6 x + log6 9 = log6 54 6
Exponential and Logarithmic Equations
Practice
DATE
10/1/09 1:55:55 PM
Chapter 3
Solve each equation.
3-4
NAME
1000
1250
1563
1
2
3
4. INTEREST RATE The effective annual
yield E for an account that is
compounded n times per year at
r percent is given by the formula
r n
E = 1 + −
n - 1. Suppose an account
pays 5%. Use a calculator to find how
many compounding periods it would take
for the effective yield to be 5.1%.
3. RICHTER SCALE The function used
to measure the magnitude R of
an earthquake is given by
R = 0.67 log (0.37E) + 1.46, where
E is the energy in kilowatt hours
that is released by the earthquake.
If the magnitude of an earthquake is 6.0,
find the approximate energy released.
b. In what week will the blog get
10,000 hits? 12th week
N = 1000(1.25)w - 1
a. Write an exponential model to find the
number of hits N in week w.
Number
of Hits
Week
2. INTERNET The table shows the number
of hits an Internet blog received for
three weeks.
005_032_PCCRMC03_893804.indd 23
Chapter 3
5
DATE
23
PERIOD
2
3
24
4
32
5
40
−t
8
t-2
−
Lesson 3-4
10/1/09 1:56:46 PM
Glencoe Precalculus
d. Using the formula in part c, find how
long it will take for the number of
bacteria to reach 5000. 39.2 hr
N = 200(2)
c. Suppose a chemical is added that
kills exactly one half of the bacteria
at the 16-hour mark. Write an
equation to find the number of
bacteria after the 16-hour mark.
N = N0(2) 8 , or N = 100(2) 8
−t
20040080016003200
16
8
1
b. Write an equation to model the
number of bacteria N at time t given
N0, the initial number of bacteria.
Population
Time (hr)
Number of
doubles
a. If there were 100 bacteria at t = 0
hours, complete the table below.
6. BIOLOGY A certain strain of bacteria
in a perfect growth medium doubles in
population every 8 hours.
radioactivity in a sample is given by the
equation ln (N) - ln (N0 ) = −kt, where
N is the current level, N0 is the original
level, k is the decay rate, and t is the
time elapsed in hours. If the decay rate
is 0.070, how many grams would be left
after 24 hours if the original amount
was 1000 grams? 186.4 grams
Exponential and Logarithmic Equations
Word Problem Practice
1. RADIOACTIVE DECAY The amount of
radium A present in a sample after t
years can be modeled by A = A0e -0.00043t,
where A0 is the initial amount. How
long will it take 50 grams to decay to
10 grams? 3743 years
3-4
NAME
Chapter 3
Enrichment
DATE
PERIOD
A11
1.36 years
24
Glencoe Precalculus
10/1/09 1:57:29 PM
Glencoe Precalculus
005_032_PCCRMC03_893804.indd 24
Chapter 3
7. Suppose the computer’s value suddenly drops because of a surplus at the end of the third
year. The company can sell the used computer for one quarter as much as it could before
the surplus. What is the new residual value of the computer? At what time does the book
value match the residual value? \$135; 5.86 years
exponential model
d. In which year is the depreciation expense zero? cannot be zero with the
c. Calculate the depreciation expense for the second year. \$600
b. If the company takes in \$10,000 in the first year, what should the earnings report
show as a balance? \$9000
a. Calculate the depreciation expense for the first year. \$1000
6. An earnings report shows the amount of money that the company takes in minus
(among other things) the depreciation expense.
over in year 3.
5. The company has decided to buy new computers every five years. Is this a wise strategy
based on this estimated useful life? Explain. No; the useful life of the machine is
4. What is the estimated useful life of the machine (in whole years)? 3 years
3.15 years
3. How long will it take for the book value to be the same as the residual value of \$500?
2. How long will it take for the book value be one-half of the asset cost?
V = 2500(0.60)t
1. Write a formula to give the book value V of the computer t years after purchase.
A computer is purchased for \$2500. Its depreciation rate is 40% per year.
The book value of an asset is equal to the asset cost minus the accumulated depreciation.
This value represents the unused amount of asset cost that the company may depreciate in
future years.
There are several methods of determining the amount of depreciation in a given year. In the
declining-balance method, the depreciation expense allowed each year is equal to the
book value of the asset at the beginning of the year times the depreciation rate.
• residual value: the expected cash value of the asset at the end of its useful life
• estimated useful life: the number of years the company can expect to use the asset
• asset cost: the amount the company paid for the asset
For tax purposes, a company distributes the cost of assets, such as equipment or a building,
as an expense over the course of a number of years. Because assets depreciate (lose some of
their value) as they get older, companies are allowed to deduct depreciation when filing
income taxes. Depreciation expense is a function of these three values.
Depreciation
3-4
NAME
Graphing Calculator Activity
DATE
PERIOD
[0, 100] scI: 10 by [0, 5] scI: 1
2. log2 (x - 1) ≤ 2 (1, 5]
3. 0 ≤ log5 x ≤ 1 (0, 5]
005_032_PCCRMC03_893804.indd 25
Chapter 3
25
will an investment double in less than 5 years? r ≥ 14.87%
ln (1 + r)
ln 2
value at a rate of r percent is given by t = −
. For what values of r
Lesson 3-4
10/1/09 2:37:41 PM
Glencoe Precalculus
5. INVESTMENT The length of time t it takes for an investment to double in
4. DEPRECIATION A family buys a car for \$25,000. If it depreciates at a
rate of 10% per year, for how long will the value of the car be above
\$15,000? up to 4.85 years
1. log3 x ≥ 2 [9, ∞)
Solve each exponential and logarithmic inequality using a table or a
graph.
Exercises
Y2 = 3
Use the intersect feature to find the point at which the logarithmic
function is equal to 3.
The graph is below the line y = 3 when x < 64.
So, log4 x ≤ 3 for (0, 64].
Y1 = −
log (X)
log (4)
Example 2
Solve log4 x ≤ 3.
Use a graph.
Use the change of base formula to enter f(x) = log4 x into a
graphing calculator. Also, enter f(x) = 3.
Use a table.
Enter the function f(x) = 150,000(1.03)x into a graphing calculator.
Use [Tblset] to make the increment 0.1 year, and use the table
feature to show the number of years and the value of the home.
So, it takes about 5.3 years for the home to be worth \$175,000.
Example 1
value increases by 3% per year. When will the value of the home
exceed \$175,000?
Let x represent the time in years. Then a function relating x and the value of
the home is f(x) = 150,000(1.03)x. To find when the value of the home will
exceed \$175,000, solve the inequality 150,000(1.03)x > 175,000.
Tables, graphs, and graphing calculators can be used to help solve
exponential and logarithmic inequalities.
Solve Exponential and Logarithmic Inequalities
3-4
NAME
Modeling with Nonlinear Regression
Study Guide and Intervention
DATE
1777.30
GDP
2501.80
1960
3771.90
1970
5161.70
1980
7112.50
1990
9817.00
2000
A12
Glencoe Precalculus
005_032_PCCRMC03_893804.indd 26
Chapter 3
26
Sample answer: The number of mice being born with
the defect has stabilized.
4. Make a conjecture about why a logistic model may fit the data.
9.639
Glencoe Precalculus
7
9.019
7.489
4
6
6.618
3
8.297
5.721
2
5
y
4.836
x
1
[0, 70] scI: 10 by [1000, 10,000] scI: 1000
[0, 10] scI: 1 by [0, 15] scI: 1
3. Find the value of the model at x = 20. f(20) = 11.94
12
f(x) = −
1 + 2e-0.3x
2. Find a logistic function to model the data.
1. Make a scatter plot of the data.
The data shows the number of mice per thousand who were
born with a certain genetic characteristic over the last 7 years.
Exercises
Step 1 Make a scatter plot.
Step 2 Find a function to model the data. The graph appears to
be rapidly rising. Try an exponential function. With the
diagnostic feature turned on and using ExpReg from the
list of regression models yields the values shown in the
[0, 70] scI: 10 by [1000, 10,000] scI: 1000
second screen. The GDP in 1950 is represented by a, and
the growth rate, 3.5% per year, is represented by 1 - b.
Notice that the correlation coefficient r ≈ 0.999 is close to 1,
indicating a close fit to the data.
Step 3 Graph the formula and the scatter plot on the
same screen. In the Y= menu, pick up this
regression equation by entering VARS ,
Statistics, EQ. The exponential fit is
nearly perfect.
Step 4 Estimate the 2020 GDP. Depending on the fit, it
should be around 19,948.
Source: U.S. Dept. of Commerce
1950
Year
Example
Use the data in the table to determine a regression
equation that best relates the gross domestic product, GDP, (in
billions of chained 2000 dollars) to the year. Estimate the 2020 GDP.
Regression
can be used to model real-world data that exhibits exponential or logarithmic
growth or decay.
PERIOD
10/1/09 10:58:51 AM
Chapter 3
Exponential, Logarithmic, and Logistic Modeling
3-5
NAME
DATE
Modeling with Nonlinear Regression
Study Guide and Intervention
(continued)
PERIOD
2
765
4
916
6
8
10
12
14
1004 1066 1115 1155 1188
765
Machines
916
1.386
1004
1.792
1066
2.079
1115
2.303
1155
2.485
1188
2.639
Brightness
(candelas)
30
8.16
40
4.59
50
60
2.04
005_032_PCCRMC03_893804.indd 27
27
Sample answer: exponential, f(x) = 33.15(0.96)x;
brightness at 100 = 0.56 candelas
Chapter 3
70
1.15
80
[0, 4] scI: 1 by [600, 1300] scI: 75
[0, 15] scI: 2 by [600, 1300] scI: 75
Lesson 3-5
10/1/09 2:37:51 PM
Glencoe Precalculus
[14, 86] sci: 1 by [-0.33, 23] sci: 5
1.50
[14, 86] scI: 1 by [-0.33, 23] scI: 1
2.94
c. Calculate an exponential or logarithmic regression
model for the data and estimate the brightness at 100 meters.
Sample answer: graph (x, ln y)
for each point in the table.
b. Create a graph that linearizes the data.
exponential
a. Make a scatter plot of the data and
describe the possible model.
20
18.36
Distance (m)
The brightness of a point of light is measured at different distances.
Exercise
Make a scatter plot of this data and draw a line through it. This
creates a straight line. Therefore, the original function is logarithmic.
y = 217x + 614
y = 217 ln x + 614
On day 9, there are y = 217 ln (9) + 614 = 1091 infected machines.
0.693
ln (day)
Step 1 Make a scatter plot of this data to get a sense of the shape.
The shape of the graph suggests a logarithmic function.
Step 2 Create a graph for (ln x, y).
Machines
Day
Example
The spread of a given computer virus is measured
over time. What model will best describe its growth? How many
machines would be infected on day 9?
• a power function y = axb, graph (ln x, ln y).
• a logarithmic function y = a ln x + b, graph (ln x, y).
• an exponential function y = abx, graph (x, ln y).
• a quadratic function y = ax2 + bx + c, graph (x, √
y ).
The correlation coefficient is a measure calculated from
a linear regression. Data can be linearized in order to measure a correlation
coefficient for nonlinear data. This is accomplished by graphing (x, r(x)),
where r(x) is the regression model you are using, such as power, exponential,
or logarithmic. If the data appears to be in a cluster about a line, then this
regression model is likely to be a good fit. To linearize data modeled by:
Linearizing Data
3-5
NAME
Chapter 3
DATE
5.1
y
11.4
34
25.6
36
57.7
38
129.7
40
291.9
42
32.4
656.8 1477.9
46
A13
[0, 50] scI: 10 by [0, 10] scI: 1
[30, 48] scI: 4 by [0, 1500] scI: 175
y
20
3.05
30
3.98
40
4.06
50
4.51
60
5.13
70
5.49
80
5.25
28
90
100
5.38
Glencoe Precalculus
10/1/09 2:38:03 PM
Glencoe Precalculus
[0, 110] scI: 10 by [0, 2] scI: 0.2
[0, 110] scI: 10 by [0, 10] scI: 1
5.11
005_032_PCCRMC03_893804.indd 28
Chapter 3
1+e
5.443776
logistic; y = −
-0.052x
d. Use the linear model to find a model for the
original data. Check its accuracy by graphing.
c. Graph the linearization and write the equation
of the line of best fit. y = 0.008x + 1.02
10 20 30 40 50 60 70 80 90 100
x
ln y 0.88 1.12 1.38 1.40 1.51 1.64 1.70 1.66 1.63 1.68
b. Linearize the data according to the given model.
a. Make a scatter plot of the data.
10
2.41
x
4. The data below is for the population of Chicago, Illinois, and Cook County, in millions,
from 1910–2000, where x is the number of years since 1900.
d. Use the linear model to find a model for the original data.
x
Check its accuracy by graphing. y = 0.0000119(1.5)
c. Graph the linearization and write the equation of
the line of best fit. y = 0.4051x - 11.340
32
34
36
38
40
42
44
46
x
ln y 1.629 2.433 3.243 4.055 4.866 5.676 6.487 7.298
b. Linearize the data according to the given model.
a. Make a scatter plot of the data.
32
3. x
19,678
c. Find the value of the model at x = 25.
c. Find the value of the model at x = 20.
[0, 12] scI: 2 by [0, 24] scI: 4
b. Find a logarithmic function to model
the data. y = 10.8 ln (x)- 2.4
44
3
4
5
6
7
8
9
9.5 12.6 15.0 17.0 18.7 20.1 21.4
a. Make a scatter plot of the data.
2
5.1
PERIOD
b. Find an exponential function to model
the data. y = 0.333e0.5493x
[0, 10] scI: 2 by [0, 30] scI: 5
a. Make a scatter plot of the data.
y
2. x
Modeling with Nonlinear Regression
Practice
1
2
3
4
5
6
7
8
y 0.58 1.00 1.73 3.00 5.20 9.00 15.59 27.00
1. x
3-5
NAME
005_032_PCCRMC03_893804.indd 29
Chapter 3
Month
Hits
2
82
4
6
8
10 12
170 265 369 480 601
2. INTERNET A popular blog has steadily
grown in the number of times the main
page is read (hits) each month.
Advertisers pay more for more popular
sites, so an accurate calculation of
projected traffic is critical. Use a scatter
plot and an exponential model to find
the number of hits at the end of the
second year. around 7000 hits
around 11.13 billion
c. Use the type of model to predict the
population of the world in 2050.
exponential
b. From the graph, what seems to be
the most appropriate model for
[0, 100] scI: 10 by [0, 6] scI: 1
a. Make a scatter plot of the data.
Source: United Nations
PERIOD
1 + 9.17e
4. BACTERIA Bacteria that is cultured in
a petri dish cannot continue to grow
forever in an exponential manner
because resources become limited. The
table below estimates the population of
bacteria in a culture, in thousands.
2007, there would have been
model. That would be about 1
rat per 36.6 people; the
estimate is very close.
b. A better estimate is that there
is a maximum of about 1 rat per
36 people. The estimated population
of New York in 2007 was about
8.3 million. How close is your model’s
230
y= −
-0.1t
a. Find a logistic model for this data.
Year 1940 1950 1960 1970 1980 1990 2000
Rats 22.6 52.6 102.7 158.0 197.0 216.6 224.9
Lesson 3-5
10/1/09 2:38:17 PM
Glencoe Precalculus
the limiting resources would
be removed and the function
would not be logistic.
c. If the culture was suddenly expanded
in size and food was continuously
added to the colony, would this model
still hold? Why or why not? No,
3,124,000
b. Estimate the population of bacteria
that will exist at hour 120.
1 + 270e
3129
y= −
-0.1x
a. Find a logistic model for the data.
Hour
12 24 36
48
60
72
84
Population 29.2 121.1 390.3 1018.3 1932.8 2642.5 2970.3
29
1930 1940 1950 1960 1970 1980 1990 2000
Year
Population 2.07 2.30 2.52 3.02 3.70 4.44 5.27 6.06
DATE
3. POPULATION A common statement is
that there is one rat per person in New
York City. Below is a fictional table with
data regarding the number of rats (in
thousands) in the city in a given year.
Word Problem Practice
1. POPULATION The table below gives the
population of the world in billions for
selected years.
3-5
NAME
Enrichment
PERIOD
10
20
30
40
50
60
70
80
90
100
3.5
3.7
3.9
4.2
4.7
5.1
5.4
A14
10
5.1
325,691
5.5
5.3
30
210,551
20
5.6
398,564
40
5.8
614,799
50
Glencoe Precalculus
005_032_PCCRMC03_893804.indd 30
Chapter 3
30
Since the data lie in a relatively straight
line after 1940, you can say that Dallas
has experienced exponential
growth from 1920 to 1990.
Years since
60
70
80
90
100
1900 x
Population y 951,527 1,327,321 1,556,390 1,852,810 2,218,899
log y
6.0
6.1
6.2
6.3
6.3
log y
Population y 135,748
Years since
1900 x
The table below shows the population of Dallas, Texas, and
the surrounding county from 1910 to 2000. Determine if an
exponential model is a good fit for the data.
Exercise
Step 3 Interpret the graph.
Since the data lie in a relatively straight line after 1940, you
can say that Las Vegas has experienced exponential growth
since 1940.
Step 2 Graph (x, log y) on the semilogarithmic paper.
log y
1
2
3
4
5
6
10
9
8
7
1
2
3
4
5
6
10
9
8
7
5.7
20
20
40
40
5.9
60
60
Glencoe Precalculus
Years
80 100 120 140
80 100 120 140
Years
6.3
Population y 3321 4859 8532 16,414 48,289 127,016 273,288 463,087 741,459 1,836333
Years since
1900 x
Step 1 Fill in the table values. Round to the nearest tenth.
Example
The table below shows the population of Las Vegas,
Nevada, and the surrounding county from 1910 to 2000. Determine if
an exponential model is a good fit for the data.
Regression equations for exponential functions may depend on creating a table
of values with x as the independent variable and log y as the dependent
variable, as shown in the example below. If you use semilogarithmic paper, you
can graph the table of values in a simplified way.
DATE
3/18/09 6:02:25 PM
Chapter 3
Graphing on Logarithmic Paper
3-5
NAME
DATE
PERIOD
005_032_PCCRMC03_893804.indd 31
Chapter 3
31
Yes, the NPV is \$868.40 greater than the cost.
b. Would this be a good investment of \$3000? Explain.
3. a. Calculate the NPV for an investment over a period of six years if the
cost of capital is 4.5% and the investment will bring a cash flow of
\$750 every year. The NPV would be about \$3868.40.
Sample answer: Yes. After one year, the gain is \$3840.05,
so they should buy the printer.
a. The last issue of the magazine sold 500 copies. If each issue of the
magazine printed in color sells 100 copies more than the previous
issue, is the printer a good investment after one year? Explain.
7924.664
1707.534
1878.287
2066.116
2272.727
-8000
(1 + r)
CF
−
n
Lesson 3-5
3/18/09 6:02:30 PM
Glencoe Precalculus
2. Four times a year, Josey and Drew publish a magazine. They want to buy
a color printer that costs \$1750. The cost of capital for this purchase
would be 6%. They are planning to raise the price of their magazine from
\$1 to \$2. Create a spreadsheet to determine the NPV for this purchase.
greater than the NPV, so it would not be cost-effective to buy
the van.
1. If the NPV is greater than the cost, the investment will pay for itself.
Based on the spreadsheet shown above, would it be cost-effective for the
company to buy the van? Explain. The cost is actually about \$75
Exercises
and r = the cost of capital, which is either the interest that will be paid on a
loan or the interest that the money would earn if it were invested.
(1 + r)
n
NPV = −
, where CFn = the cash flow in period n
n
CF
When a business owner is considering
Period (n)
CF
a major purchase, it is important to determine
whether the investment will be profitable in the
0
-8000
future. Consider the example of a local
1
2500
restaurant-delivery service that is debating whether
2
2500
to buy a van for \$8000. The owners of the company
estimate that using the van will generate \$2500
3
2500
per year over four years. They can use the
4
2500
following formula to find the present value of the
Total
future cash flow to find the Net Present Value (NPV),
NPV - Cost
-75.3364
that is, how much the profits would be worth in
Cost of Capital (r)
10%
today’s dollars.
Net Present Value
3-5
NAME
Quiz 1
(Lessons 3-1 and 3-2)
Page 33
1.
81
D
16
1.
A
2.
G
3.
D
4.
J
5.
A
9
3.
4.
Page 35
B
2.
3.
Mid-Chapter Test
7.3 yr
1
= -4
log 3 −
4.
(Lesson 3-4)
Page 34
1. \$16,976.03
2.
Quiz 3
0.46
y
2
5.
12
8
4
-2 0
5.
2
4
6x
Quiz 4 (Lesson 3-5)
y
Page 34
x
1.
Quiz 2 (Lesson 3-3)
Page 33
1.
2.289
2.
0.8
3.
6
2.
D
ln (1 - p) + ln d 1
−
ln (2 + 5r)
5. 6
Chapter 3
y = 0.24 (2.34) x
3. 215.7
6.
ln (x + 1) 1
−
ln (x - 5)
4
2x + y
log3 −
3
4.
4.
B
0
exponential;
y = 15.198 (1.07) x
y = 0.0058x +
5. 3.726
7.
x-y
√
8.
295
No; she will
have only
9. \$1760.27.
10.
A15
17,239
Glencoe Precalculus
Vocabulary Test
Page 36
Form 1
Page 37
1.
transcendental
1. function
2.
Page 38
A
D
12.
H
13.
D
14.
F
15.
C
16.
G
17.
A
18.
G
H
2. algebraic function
3.
11.
exponential function
3.
A
logistic growth
4. function
5. common logarithm
6. natural logarithm
4.
J
5.
B
6.
J
7.
B
8.
G
9.
C
20.
G
10.
J
B:
D
7. linearize
8. natural base
19.
A
9. an account with no
waiting period
between interest
payments
10. the inverse of the
function f(x) = b
Chapter 3
x
A16
Glencoe Precalculus
Form 2A
Page 39
1.
2.
3.
B
Page 40
11.
D
12.
Page 42
11.
A
H
12.
F
13.
A
13.
A
14.
F
14.
J
15.
B
15.
B
16.
H
16.
H
17.
A
18.
H
19.
B
20.
J
7 log 7 2x
H
C
Form 2B
Page 41
1.
2.
B
J
3.
C
4.
F
4.
F
5.
C
5.
B
6.
H
6.
H
7.
D
8.
F
9.
B
10.
J
Chapter 3
18.
J
19. A
F
20.
B:
11 log 11x
7.
D
8.
G
9.
D
10.
A17
G
17. A
B:
Glencoe Precalculus
Form 2C
Page 43
Page 44
y
1.
8
exponential
12.
3.14
13.
1.15
x
0
2.
11.
y
4
−8
4
−4 0
3
14.
−4
−8
15.
log913 + 2 log9 x
- 5 log 9 y
\$1987.87
3.
2.49 g
4.
5.
x2
log −
8x
log −1 1296 = -4
6
3
-−
4
6.
13
17.
3
shifted 2 units right,
expanded vertically
by a factor of 3, and
18.reflected in the x-axis
5
7.
0
9.
2.776
10.
-7.432
8.66%
19.
y
8.
Chapter 3
16.
x
In y = 0.0582x +
4.4657
20.
3
−
= loga x
B:
A18
2
Glencoe Precalculus
Form 2D
Page 45
Page 46
1.
y
x
0
11.
logistic
12.
3.138
13.
1.148
y
2.
14.
x
0
15.
16.
3 log13 2 +
10 log13 x - log13 y
4
−
3
\$11,778.72
3
log16 8 = −
4
6.
-3
7.
4
8
shifted 2 units right,
expanded vertically
by a factor of 3, and
18. reflected in the x-axis
19.
11.18 years
y
8.
0
x
ln y = 0.0774x +
20. 3.8065
9.
10.
Chapter 3
4.120
-0.791
B:
A19
a2
logb −
c
Glencoe Precalculus
17.
5.
3
y2
√
395
3.
4.
ln 3
Form 3
Page 47
Page 48
1.
y
x
0
15.9
11.
10
−
3
12.
2.
13.
y
2
−8−6−4 0
2 4 6 8x
In 5 + 1.5
In x - y In 2
14.
5
y4
√
log −
−4
−6
−8
15.
9
()
3
4
16. -0.2877, 0
1696
3.
17.
\$18,281.88
4.
5.
18.
logy 7 = x - 3
6.
-2
7.
6
8.
y
0
19.
3.889
10.
3.182
Chapter 3
g(x) = -3(4) x - 2
7.296 years
x
20.
9.
0
y = 32.07(1.065)x
In 3.12
−
≈ 0.38 and
B:
A20
3
In 0.49
−
≈ -0.24
3
Glencoe Precalculus
Page 49, Extended-Response Test
1e. For x < 1, log5 x > log3 x.
For x > 1, log5 x < log3 x.
For x = 1, log5 x = log3 x.
They intersect at (1, 0).
y
1a.
y = 3x
y
x
0
y = log5 x
1b. For x > 0, 5x > 3x. For x < 0, 5x < 3x. For
x = 0, 5x = 3x. They intersect at (0, 1).
x
0
y
1f. The graph of g(x) = log8 x is the graph of
f(x) = log2 x compressed vertically by a
y = 5x
1
. By the Change of Base
factor of −
3
log2 x
1
, or −
log2 x.
Formula, log8 x = −
x
0
log2 8
(3)
1
1c. The graph of f(x) = −
x
3
x
is the reflection
2. Sample answer: A colony of 9 bacteria
doubles every minute. When will the
population of the colony be 178?
of g(x) = 3 in the y-axis. They intersect
at (0, 1).
178 = 9 2x
y
log 178 - log 9
log 2
x = − or about 4.3
⎞x
y = ⎛⎝ 1 ⎠
3
The colony will number 178 in about
4.3 minutes.
x
0
3. log2 (x + 3) + log2 (x - 2) = 3
1d. The graph of g(x) = log3 x is the
reflection of f(x) = 3x in the line y = x.
The graphs do not intersect.
log2 [(x + 3)(x - 2)] = 3
(x + 3)(x - 2) = 23
2
x +x-6=8
y
Product Property
Definition of logarithm
Multiply.
2
x + x - 14 = 0
-1 ± √1
+ 56
2
-1 ± √
57
=−
2
-1 - √
57
The solution x = − is
2
f(x)
x= −
0
g(x )
x
extraneous because it makes both
logarithms undefined.
(continued on the next page)
Chapter 3
A21
Glencoe Precalculus
log 178 = log 9 + x log 2
Page 49, Extended-Response Test Sample Answers (continued)
e2x - 3ex ≤ -2
4.
Original inequality
e2x - 3ex + 2 ≤ 0
x
x
(e - 1)(e - 2) ≤ 0
Factor.
Find the critical numbers.
ex - 1 = 0
ex - 2 = 0
ex = 1
ex = 2
x
x
ln e = ln 1
ln e = ln 2
x ln 1 = ln 1
x ln 1 = ln 2
x=0
Set each factor equal to 0.
Solve.
One-to-One Property
Inverse Property
x = ln 2
≈ 0.693
Create a sign chart.
Test Y = 0.5
+-
Test Y = -1
++
0
Test Y = 2
++
ln 2
(0.693)
Y
Because f(x) is negative in the middle
interval, the solution of
e2x - 3x2 ≤ -2 is [0, ln 2].
Chapter 3
A22
Glencoe Precalculus
Standardized Test Practice
Page 50
1.
A
B
C
D
2.
F
G
H
J
3.
4.
F
A
B
G
B
C
H
C
D
J
9.
A
10.
B
C
D
F
G
H
J
11.
A
B
C
D
12.
F
G
H
J
13.
A
B
C
D
14.
F
G
H
J
15.
A
B
C
D
16.
F
G
H
J
17.
A
B
C
D
18.
F
G
H
J
D
5.
A
Page 51
6.
F
7.
A
8.
F
G
B
G
Chapter 3
H
C
H
J
D
J
A23
Glencoe Precalculus
Standardized Test Practice
(continued)
Page 52
19.
(f ◦ g) (x) =
4
−
x2 - 2x +1
-4
-2, -1, 1.5
20.
removable
21. discontinuity
y
22.
x
0
24.
5000
-18
25.
23.
y = 2.497x + 177.88
26.
27. y = 2029
y
28a.
0
x
D = (-4, ∞);
b. R = (-∞, ∞)
asymptote at x = -4;
end behavior
lim f(x) = -∞,
+
x→-4
c.
lim f(x) = ∞
x→∞
d. increasing on (-4, ∞)
Chapter 3
A24
Glencoe Precalculus
```