Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1)
ln x
yz
=
1)
A)
ln x+1
2(ln y– ln z)
B)
ln x–1
2(ln y+ ln z)
C)
ln x–ln y +ln z
D)
ln x
ln y ln z
E)
ln x–1
2(ln y– ln z)
2)
The population of a city is given by P= 1,000,000(1.02)t where t is the number of years after 1987.
The population in 1989 was
2)
A)
1,020,000.
B)
1,040,400.
C)
1,002,000.
D)
1,040,000.
E)
1,004,000.
3)
Initially, three are 50 milligrams of a radioactive substance. The substance decays according to the
equation N= 50e–0.01t, where N is the number of milligrams present after t hours. The number of
hours it takes for 10 milligrams to remain is
3)
A)
–ln(0.01)
5.
B)
1
5
ln(–0.01) .
C)
50e–0.1.
D)
–
1
5
ln(0.01) .
E)
–
ln 1
5
0.01 .
4)
ln (xy)2
z3=
4)
A)
ln 2 + ln x+ ln y– ln 3 – ln z
B)
ln 2 ln x ln y– ln 3 ln z
C)
ln(xy)2
ln z3
D)
2(ln x+ ln y) – 3 ln z
E)
2 ln x ln y– 3 ln z
5)
If a radioactive substance decays according to the equation N=40e–0.03t, where N is the number of
milligrams present after t days, then the half–life, in days, of the substance is given by
5)
A)
ln 2
0.03 .
B)
–0.03
ln 2 .
C)
–ln 2
0.03 .
D)
0.03
ln 2 .
E)
ln 2
40 .
6)
log51
25 =
6)
A)
–5
B)
–2
C)
–1
2
D)
1
2
E)
–4
7)
If log 2 = 0.3010 and log 3 = 0.4771, then log2 3 =
7)
A)
0.3010
0.4771
B)
0.4471 – 0.3010
C)
0.4771
0.3010
D)
(0.3010)(0.4771)
E)
0.3010 – 0.4771
8)
If 10logx= 5, then x =
8)
A)
5
2.
B)
log 5.
C)
1.
D)
0.
E)
5.
9)
If log 3 = 0.4771, then log 0.3 =
9)
A)
–0.5229.
B)
0.5229.
C)
–4.771.
D)
–1.4771.
E)
1.4441.
10)
If log4(x+ 6) = 2 –log4 x, then x=
10)
A)
–4.
B)
10.
C)
8.
D)
4.
E)
2.
11)
log6
564=
11)
A)
–5
4
B)
5
4
C)
–4
5
D)
0
E)
4
5
12)
If $1000 is invested for 2 years at 6% compounded quarterly, then the compound amount at the end
of the period is
12)
A)
$1593.84.
B)
$1120.00.
C)
$1141.23.
D)
$1126.49.
E)
$1123.60.
13)
The above graph is best represented by
13)
A)
y =4x
B)
y
=1
4
x
C)
y
= ln 4
D)
y=e4
E)
log4x
14)
Writing 1
6 [ln x– 2(ln y+ 2 ln z)] as a single logarithm gives
14)
A)
ln
6x
y2z4.
B)
ln 6x
y2z2.
C)
ln
6x
yz2.
D)
ln 6xz4
y2.
E)
ln 6x
y2z4.
15)
If 22x+1=8x–3, then x=
15)
A)
log2 3.
B)
6.
C)
–7.
D)
10.
E)
–4.
16)
log 10,000 – 6 log 10 =
16)
A)
0
B)
–6log 10
C)
1
D)
10,000
610
E)
4
17)
ln 6 + ln 1
6+ ln e2=
17)
A)
0
B)
1
C)
2
D)
3
E)
4
18)
ln e– ln 1 =
18)
A)
6
B)
ee–1
C)
1
D)
0
E)
n
(
e
– 1)
19)
If log 2 = 0.3010 and log 3 = 0.4771, then log 27
2=
19)
A)
04.283.
B)
0.1761.
C)
1.1303.
D)
1.7323.
E)
3.1761.
20)
If log(x+3)4= 4, then x can equal
20)
A)
44.
B)
1
10 .
C)
–3.
D)
12.
E)
7.
21)
Writing 2 ln x–1
3ln y+ 4 ln z as a single logarithm gives
21)
A)
ln x2z4
3y
.
B)
ln(x2–
3y+z4).
C)
ln x2z4
y3.
D)
ln 2x–1
3y+ 4z.
E)
ln(x2+z4)
ln 3y
.
22)
If $500 is invested for 3 years at 7% compounded semiannually, then the compound interest at the
end of the period is
22)
A)
$150.36.
B)
$112.52.
C)
$114.63.
D)
$122.56.
E)
$120.37.
23)
If $10,000 is invested at 16% compounded quarterly, then the compound amount at the end of six
years is
23)
A)
$25,633.04.
B)
$21,173.75.
C)
$26,678.42.
D)
$26,987.33.
E)
$24,278.09.
24)
If ln e5x–4= 6, then x=
24)
A)
4
5.
B)
ln 4
5.
C)
1.
D)
2.
E)
e4/5.
25)
If e ln(4x+3) = 7, then x=
25)
A)
0.
B)
1.
C)
–2.
D)
1
4 ln 7
3.
E)
2.
26)
Changing log7(4x) to natural logarithms gives
26)
A)
ln (4
x
).
B)
ln
e
ln(4
x
).
C)
ln 7
ln(4
x
).
D)
ln(4
x
)
ln
e
.
E)
ln(4
x
)
ln 7 .
27)
If ln x+ ln 2 = ln 5, then x=
27)
A)
5
2.
B)
5.
C)
3.
D)
2
5.
E)
1
3.
28)
If log2(4x + 1) = 3, then x=
28)
A)
11
4
B)
7
4
C)
3
4
D)
9
4
E)
5
4
29)
If log 2 = 0.3010 and log 3 = 0.4771, then log 8
3=
29)
A)
2.8239.
B)
1.3801.
C)
1.8927.
D)
0.2136.
E)
0.4259.
30)
If p=54–q, then in terms of common logarithms, q=
30)
A)
p
– 4 log 5.
B)
log54–p.
C)
5 –log 5
log
p
.
D)
4 –log
p
log 5 .
E)
5 –log
p
log 4 .
31)
Of the following the best approximation of e is
31)
A)
1.8.
B)
1.4.
C)
2.3.
D)
2.7.
E)
3.1.
32)
If log2(x + 3) = – 2, then x=
32)
A)
–1
B)
–4
3
C)
3
4
D)
–7
E)
–11
4
33)
log 0.001 =
33)
A)
–1
B)
–2
C)
–3
D)
–4
E)
–5
34)
If log 2 = 0.3010 and log 3 = 0.4771, then log 18 =
34)
A)
1.9030.
B)
0.1761.
C)
1.2552.
D)
0.0685.
E)
2.3343.
35)
The above graph is best represented by
35)
A)
y=4x
B)
y
=1
4
x
C)
y=log4
D)
y=e4
E)
y
= ln 4
36)
log 300
100 =
36)
A)
log 3
B)
3
C)
log 200
D)
2 + log3
2
E)
none of the above
37)
If 2e3x– 5 = 3, then x=
37)
A)
3
ln 4
B)
1
3ln 7
2
C)
4
3
D)
ln 4
3
E)
ln 3
4
D
38)
The value of log3 0 is
38)
A)
0
B)
–1
C)
1
D)
3
E)
not defined
E
39)
If log4x= – 3, then x=
39)
A)
64
B)
81
C)
1
81
D)
–12
E)
1
64
E
A
40)
Which of the following is true? If f(x) = ln x, then
40)
A)
both the domain and the range of f are all real numbers.
B)
the domain of f is all real numbers and the range is all positive real numbers.
C)
both the domain and the range of f are all positive real numbers.
D)
the domain of f is all real numbers except zero and the range is all real numbers.
E)
the domain of f is all positive real numbers and the range is all real numbers.
41)
log5 125 =
41)
A)
0
B)
1
C)
2
D)
3
E)
4
42)
Which of the following is true? If f(x) =ex, then
42)
A)
the domain of f is all positive real numbers and the range is all real numbers.
B)
both the domain and range of f are all positive real numbers.
C)
the domain of f is all real numbers and the range is all positive real numbers.
D)
both the domain and range of f are all real numbers.
E)
the domain of f is all real numbers except zero and the range is all real numbers.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
43)
Find x and express your answer in terms of common logarithms: 4x= 3
43)
44)
How long will it take for $100 to amount to $200 at an interest rate of 10% compounded
annually? Give your answer to 2 decimal places.
44)
45)
Find x: logx16 = 4
45)
46)
Find x: log 0.01 =x
46)
47)
Assume that log 6 = 0.7782. Determine the value of log(0.6).
47)
48)
Graph f(x) = 3 2
3
x
48)
13
49)
Consider h(x) = 2ln(x) – 3
(a) Graph h(x)
(b) Find the domain of h(x)
49)
50)
Use a graphing calculator to approximate the solution 2x–3x= 20.
50)
51)
If $2000 is invested for 3 years at 8% compounded quarterly, find
(a) the compound amount and
(b) the compound interest.
51)
52)
Solve: 10log x3= 27
52)
53)
Find x and express your answer in terms of natural logarithms: e4x= 2
53)
54)
Steve wants to use his graphing calculator to check his sketch of y=log5x but his
calculator does not make log5 calculations. Find two equations he could use.
54)
55)
Find x: logx(6 – 4x–x2) = 2
55)
56)
A radioactive substance decays according to the equation N=10e–0.04t, where N is the
number of milligrams present after t days. Find the half–life of the substance.
56)
57)
Evaluate and simplify: log5
352
57)
58)
Assume the amount of paper being recycled quadruples every year.
(a) Make a table of the factor of increase in the amount of paper being recycled at each
year for 0 to 3 years. For each year, write an expression for the factor of increase as a
power of some base.
(b) What base did you use? How does that base relate to the problem?
(c) Use your table to graph the factor of increase as a function of years.
(d) Use your graph to estimate when the recycling will have grown by a factor
of 50.
58)
15
59)
Assume your $2000 investment,which is guaranteed to triple every decade, has a one time
$10 service charge.
(a) Make a table of the value of your investment without the service charge at each decade
from 0 to 3.
(b) Make a table of the value of your investment with the service charge deducted at each
decade from 0 to 3.
(c) How could you use a graph of (a) to make a graph of (b)?
59)
60)
The demand function for a product is p= 45 5–q/10 where q is the number of units and p is
the price of one unit. At what price will the demand be 5 units? How many units will be
demanded if the price is $12.42?
60)
61)
Graph y=f(x) =3x.
61)
62)
Solve for x: 42x= 2
62)
63)
April wants to use her graphing calculator to check her sketch of y=log8x but her
calculator does not make log8 calculations. Find two equations she could use.
63)
64)
Find x: log x= 3
64)
65)
Find x: log x= – 2
65)
66)
Solve for x: logx(2x+ 3) = 2
66)
17
67)
A graphical look at Bacteria Growth: If 100 bacteria are present at the start, the number of
bacteria in a culture which changes by constant factor f every hour is given by N(t) = 100(
ft). Use a graphing calculator to graph this function for various values of f where f> 0.
Describe how the graphs where 0 <f< 1 differ from the graphs where f> 1. How does the
number of bacteria change when 0 <f< 1? How does the number of bacteria change when
f> 1? Describe the graph where f= 1. How does the number of bacteria change when f= 1?
67)
68)
Simplify: 105log x
68)
69)
The sales manager of a department store finds that its daily sales begins to fall after the end
of a promotional campaign. The sales in dollars as a function of the number of days after
the campaign’s end is given by S(d) = 36,000 9
8
–0.1d. If she does not want sales to drop
below 30,000 per day before starting a new campaign, when should she start a new
campaign?
69)
70)
Solve for x: (2 +x)5= 129.3
70)
71)
Suppose that the number of patients admitted into a hospital emergency room during a
certain hour of the day has a Poisson distribution with mean 3. Find the probability that
during that hour there will be exactly two emergency patients. Assume that e–3= 0.05.
71)
72)
Write the following in terms of ln(x+ 2) and ln(x+ 4): ln (x+2)2(x+ 4)
72)
18
Explanation:
73)
The multiplicative decrease in purchasing power P after t years of inflation at 3% can be
modeled by
P=e–0.03t. Graph the decrease in purchasing power as a function of t years.
73)
74)
Evaluate and simplify: ln e3
74)
75)
Write log2(x+ 4) in terms of common logarithms.
75)
76)
Evaluate and simplify: ln e
76)
77)
If an earthquake is 103(x–1) times as intense as a zero–level earthquake, what is its
measurement on the Richter Scale? Write as logarithmic expression and simplify.
77)
19
Explanation:
78)
The number of years it takes for an amount which is invested at an annual rate of p and
compounded continuously to become m times as large is given by t=ln(m)
p. How long
does it take an investment to triple if it is invested at an annual rate of 8% and
compounded continuously?
78)
79)
Solve for x: 3log3x+log34= 8
79)
80)
Find the equations of the graph that is obtained from the graph of y=ex shifted
(a) 3 units down;
(b) 3 units to the right.
80)
81)
Find x: log4 2 =x
81)
82)
The magnitude (Richter Scale) of an earthquake is given by R= log I
I0 where I is the
intensity of the earthquake and I0 is the intensity of a zero–level reference earthquake. I
I0
represents how many times greater than the earthquake is than the reference earthquake. If
an earthquake measured 7.5 on the Richter Scale, how many more times intense is it than a
barely–felt earthquake?
82)
83)
How much more on the Richter Scale is an earthquake with intensity 300,000 times the
intensity of a zero–level earthquake than an earthquake with intensity 150,000 times the
intensity of a zero–level earthquake? Write as an expression involving logarithms. Simplify
by combining logarithms and then use a calculator to evaluate.
83)
84)
Solve for x: e2ln(2x)= 4
84)