Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
The graph of an exponential function is given. Select the function for the graph from the functions listed.
1)
1)
A)
f(x) =3x+ 2
B)
f(x) =3x
C)
f(x) =3x– 2
D)
f(x) =3x – 2
2)
2)
A)
f(x) =5–x
B)
f(x) = – 5x
C)
f(x) = – 5–x
D)
f(x) =5x
The graph of a logarithmic function is given. Select the function for the graph from the options.
3)
3)
A)
f(x) = 1 –log 4x
B)
f(x) =log 4x
C)
f(x) =log 4(–x)
D)
f(x) = – log 4x
The graph of an exponential function is given. Select the function for the graph from the functions listed.
4)
4)
A)
f(x) = – 2–x
B)
f(x) = – 2x
C)
f(x) =2x
D)
f(x) =2–x
The graph of a logarithmic function is given. Select the function for the graph from the options.
5)
5)
A)
f(x) =log 3x + 2
B)
f(x) =log 3x
C)
f(x) =log 3(x + 2)
D)
f(x) =log 3(x – 2)
6)
6)
A)
f(x) =log 5(x – 1)
B)
f(x) =log 5(x + 1)
C)
f(x) =log 5x
D)
f(x) =log 5x + 1
7)
7)
A)
f(x) =log 5(–x)
B)
f(x) =log 5x
C)
f(x) = – log 5x
D)
f(x) = 1 –log 5x
The graph of an exponential function is given. Select the function for the graph from the functions listed.
8)
8)
A)
f(x) =3x
B)
f(x) =3x – 1
C)
f(x) =3x– 1
D)
f(x) =3x+ 1
9)
9)
A)
f(x) =4–x
B)
f(x) = – 4x
C)
f(x) = – 4–x
D)
f(x) =4x
The graph of a logarithmic function is given. Select the function for the graph from the options.
10)
10)
A)
f(x) =log 4x – 2
B)
f(x) =log 4(x – 2)
C)
f(x) =log 4x
D)
f(x) =log 4(x + 2)
11)
11)
A)
f(x) = – log 5x
B)
f(x) = 1 –log 5x
C)
f(x) =log 5(–x)
D)
f(x) =log 5x
The graph of an exponential function is given. Select the function for the graph from the functions listed.
12)
12)
A)
f(x) =2x
B)
f(x) =2x– 1
C)
f(x) =2x+ 1
D)
f(x) =2x + 1
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Present data in the form of tables. For the data set shown by the table,
a. Create a scatter plot for the data.
b. Use the scatter plot to determine whether an exponential function or a logarithmic function is the best
choice for modeling the data.
13)
Number of Homes Built in a Town by Year
Year Number of Homes
1985 11
1991 91
1994 146
1997 192
2002 224
13)
7
14)
Percentage of Population Living in the
South Suburbs of a Large City
Year Percent
1950 55
1960 69
1970 73
1980 75
2000 77
14)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Graph the function.
8
15)
Use the graph of f(x) =ex to obtain the graph of g(x) =1
2ex.
15)
A)
B)
C)
D)
Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic
expressions. Give the exact answer.
16)
log 4x = log 5+ log (x – 1)
16)
A)
4
3
B)
–5
9
C)
{–5}
D)
{5}
Solve the problem.
17)
The population in a particular country is growing at the rate of 1.2% per year. If 10,184,000 people
lived there in 1999, how many will there be in the year 2007? Use y =yoe0.012t and round to the
nearest ten–thousand.
17)
A)
12,330,000
B)
13,450,000
C)
10,990,000
D)
11,210,000
Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose
coefficient is 1. Where possible, evaluate logarithmic expressions.
18)
ln x +7ln y
18)
A)
ln x
y7
B)
ln xy7
C)
ln 7xy
D)
ln (x +7y)
Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
19)
log 7(7x)
19)
A)
1 +log 7x
B)
1
C)
x
D)
7
Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose
coefficient is 1. Where possible, evaluate logarithmic expressions.
20)
(log am–log an) + 4 log ak
20)
A)
log a
m
k4n
B)
log amk4n
C)
log a
4mk
n
D)
log a
mk4
n
21)
9logby +6logbz
21)
A)
logb(yz)15
B)
54logbyz
C)
logby9z6
D)
15logbyz
Solve the problem.
22)
The pH of a solution ranges from 0 to 14. An acid has a pH less than 7. Pure water is neutral and
has a pH of 7. The pH of a solution is given by pH = – log x where x represents the concentration of
the hydrogen ions in the solution in moles per liter. Find the hydrogen ion concentration if the
pH =3.
22)
A)
10–3
B)
1.1
C)
103
D)
0.33
Write the equation in its equivalent exponential form.
23)
log 3 x =2
23)
A)
3x=2
B)
32= x
C)
x2=3
D)
23= x
Approximate the number using a calculator. Round your answer to three decimal places.
24)
2.4
24)
A)
15.648
B)
36.462
C)
15.601
D)
7.540
11
Graph the function.
25)
Use the graph of f(x) = ln x to obtain the graph of g(x) =4– ln x.
25)
A)
B)
C)
D)
Solve the equation by expressing each side as a power of the same base and then equating exponents.
26)
3(6 – 3x) =1
27
26)
A)
1
9
B)
{3}
C)
{–3}
D)
{9}
Solve the problem.
27)
The function A =A0e–0.0077x models the amount in pounds of a particular radioactive material
stored in a concrete vault, where x is the number of years since the material was put into the vault.
If 400 pounds of the material are initially put into the vault, how many pounds will be left after 80
years?
27)
A)
216 pounds
B)
225 pounds
C)
183 pounds
D)
178 pounds
Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic
expressions. Give the exact answer.
28)
log 6x2=log 6(4x + 32)
28)
A)
B)
4
3
C)
{8, –4}
D)
{8}
Graph the function.
29)
Use the graph of f(x) =4x to obtain the graph of g(x) =4–x.
29)
13
A)
B)
C)
D)
Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic
expressions. Give the exact answer.
30)
log 3x = log 4+ log (x – 1)
30)
A)
{–4}
B)
{4}
C)
–4
7
D)
3
2
14
Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
31)
log x
10
31)
A)
10x
B)
–10x
C)
log x – 1
D)
log x +1
Solve the equation by expressing each side as a power of the same base and then equating exponents.
32)
2(3x + 7)=1
4
32)
A)
{1}
B)
{3}
C)
{–3}
D)
1
2
Solve the problem.
33)
The size of the bear population at a national park increases at the rate of 4.3% per year. If the size of
the current population is 108, find how many bears there should be in 5 years. Use y =yoe0.043t
and round to the nearest whole number.
33)
A)
136
B)
134
C)
138
D)
132
34)
Larry has $2900 to invest and needs $3500 in 11 years. What annual rate of return will he need to
get in order to accomplish his goal, if interest is compounded continuously? (Round your answer to
two decimals.)
34)
A)
2.55%
B)
3.55%
C)
1.55%
D)
1.71%
Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose
coefficient is 1. Where possible, evaluate logarithmic expressions.
35)
3log42+1
6log4 (r –3) –1
2log4 r
35)
A)
log41
4
r –3
r
B)
log48r –3
12r
C)
log43r –9
12r
D)
log486r –3
r
Solve the problem.
36)
Find out how long it takes a $3000 investment to earn $400 interest if it is invested at 8%
compounded quarterly. Round to the nearest tenth of a year. Use the formula A = P 1 +r
n
nt.
36)
A)
2 years
B)
1.4 years
C)
1.8 years
D)
1.6 years
Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic
expressions. Give the exact answer.
37)
log (2+ x) – log (x – 2) = log 3
37)
A)
{4}
B)
{–4}
C)
3
2
D)
Evaluate the expression without using a calculator.
38)
ln e14x
38)
A)
14x
B)
1
14x
C)
–14x
D)
e14x
Solve the problem.
39)
The function f(x) = 1 +1.5 ln (x + 1) models the average number of free–throws a basketball player
can make consecutively during practice as a function of time, where x is the number of consecutive
days the basketball player has practiced for two hours. After how many days of practice can the
basketball player make an average of 9 consecutive free throws?
39)
A)
785 days
B)
206 days
C)
208 days
D)
787 days
40)
The size of the raccoon population at a national park increases at the rate of 4.3% per year. If the
size of the current population is 157, find how many raccoons there should be in 4 years. Use the
function f(x) =157e0.043t and round to the nearest whole number.
40)
A)
186
B)
190
C)
188
D)
184
Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose
coefficient is 1. Where possible, evaluate logarithmic expressions.
41)
log 3(x + 3) –log 3(x + 6)
41)
A)
log 3–3
B)
log 3
x + 3
x – 6
C)
log 3(x2+ 9x + 18)
D)
log 3
x + 3
x + 6
Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic
expressions. Give the exact answer.
42)
8 ln (4x) =56
42)
A)
e7/4
B)
e7
4
C)
e7
D)
7
ln 4
17
Solve the exponential equation. Express the solution set in terms of natural logarithms.
43)
4x + 4 =52x + 5
43)
A)
ln 55
44–4
52
B)
{7 ln 5 – 5 ln 4}
C)
5 ln 5 – 4 ln 4
ln 4 – 2 ln 5
D)
{ln 5 – ln 4}
Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
44)
log 2
x2
y7
44)
A)
7log 2y –2log 2x
B)
2log 2x +7log 2y
C)
2
7log 2(x
y)
D)
2log 2x –7log 2y
Solve.
45)
The value of a particular investment follows a pattern of exponential growth. In the year 2000, you
invested money in a money market account. The value of your investment t years after 2000 is
given by the exponential growth model A =7900e0.056t. When will the account be worth $12,365?
45)
A)
2007
B)
2009
C)
2010
D)
2008
Solve the exponential equation. Use a calculator to obtain a decimal approximation, correct to two decimal places, for the
solution.
46)
8ex=13
46)
A)
0.49
B)
–0.49
C)
0.21
D)
–0.21
Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic
expressions. Give the exact answer.
47)
7+3 ln x =15
47)
A)
ln 8
3
B)
e8
3
C)
8
3 ln 1
D)
e8/3
Write the equation in its equivalent logarithmic form.
48)
b3=2744
48)
A)
logb3=2744
B)
log2744 b=3
C)
logb2744 =3
D)
log32744 =b
Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
49)
logb
xy7
z2
49)
A)
logbx +7logby –2logbz
B)
logbx +logby7–logbz2
C)
logbx +7logby +2logbz
D)
logbx +logby7+logbz2
Solve the problem.
50)
The population of a certain country is growing at a rate of 1.4% per year. How long will it take for
this country’s population to double? Use the formula t =ln 2
k, which gives the time, t, for a
population with growth rate k, to double. (Round to the nearest whole year.)
50)
A)
49 years
B)
50 years
C)
51 years
D)
52 years
Approximate the number using a calculator. Round your answer to three decimal places.
51)
e3.4
51)
A)
29.964
B)
27.843
C)
9.242
D)
30.264
Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose
coefficient is 1. Where possible, evaluate logarithmic expressions.
52)
3log x2+log x3
52)
A)
log x18
B)
log x24
C)
log x2
D)
3log x6
Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
53)
log 6
x – 6
x8
53)
A)
log 6(x – 6) +8log 6x
B)
8log 6x –log 6(x – 6)
C)
log 6(x – 6) –8log 6x
D)
log 6(x – 6) –log 6x
Solve the exponential equation. Use a calculator to obtain a decimal approximation, correct to two decimal places, for the
solution.
54)
4x=19
54)
A)
0.77
B)
0.47
C)
2.12
D)
4.08
Graph the function.
20