Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1)
If ln e5x–4= 6, then x=
1)
A)
1.
B)
2.
C)
ln 4
5.
D)
e4/5.
E)
4
5.
2)
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
2)
A)
ln 2
40 .
B)
0.03
ln 2 .
C)
–ln 2
0.03 .
D)
–0.03
ln 2 .
E)
ln 2
0.03 .
3)
Which of the following is true? If f(x) =ex, then
3)
A)
the domain of f is all real numbers except zero and the range is all real numbers.
B)
the domain of f is all positive real numbers and the range is all 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)
both the domain and range of f are all positive real numbers.
4)
log5 125 =
4)
A)
0
B)
1
C)
2
D)
3
E)
4
5)
If log4x= – 3, then x=
5)
A)
64
B)
1
64
C)
1
81
D)
–12
E)
81
6)
ln (xy)2
z3=
6)
A)
2 ln x ln y– 3 ln z
B)
ln(xy)2
ln z3
C)
ln 2 + ln x+ ln y– ln 3 – ln z
D)
2(ln x+ ln y) – 3 ln z
E)
ln 2 ln x ln y– ln 3 ln z
7)
The above graph is best represented by
7)
A)
y=4x
B)
y=log4
C)
y=e4
D)
y
=1
4
x
E)
y
= ln 4
8)
If 22x+1=8x–3, then x=
8)
A)
log2 3.
B)
–4.
C)
6.
D)
10.
E)
–7.
9)
ln e– ln 1 =
9)
A)
0
B)
1
C)
n
(
e
– 1)
D)
6
E)
ee–1
10)
If p=54–q, then in terms of common logarithms, q=
10)
A)
p
– 4 log 5.
B)
5 –log 5
log
p
.
C)
log54–p.
D)
4 –log
p
log 5 .
E)
5 –log
p
log 4 .
11)
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
11)
A)
1,004,000.
B)
1,040,400.
C)
1,002,000.
D)
1,020,000.
E)
1,040,000.
12)
If log2(4x + 1) = 3, then x=
12)
A)
11
4
B)
3
4
C)
9
4
D)
5
4
E)
7
4
3
13)
log 0.001 =
13)
A)
–1
B)
–2
C)
–3
D)
–4
E)
–5
14)
If ln x+ ln 2 = ln 5, then x=
14)
A)
3.
B)
5
2.
C)
5.
D)
2
5.
E)
1
3.
15)
The value of log3 0 is
15)
A)
0
B)
–1
C)
1
D)
3
E)
not defined
C
16)
Writing 1
6 [ln x– 2(ln y+ 2 ln z)] as a single logarithm gives
16)
A)
ln 6xz4
y2.
B)
ln
6x
y2z4.
C)
ln 6x
y2z2.
D)
ln
6x
yz2.
E)
ln 6x
y2z4.
17)
If log 2 = 0.3010 and log 3 = 0.4771, then log 8
3=
17)
A)
2.8239.
B)
1.3801.
C)
1.8927.
D)
0.2136.
E)
0.4259.
18)
If 10logx= 5, then x =
18)
A)
0.
B)
5
2.
C)
log 5.
D)
1.
E)
5.
19)
If log 2 = 0.3010 and log 3 = 0.4771, then log2 3 =
19)
A)
0.4771
0.3010
B)
(0.3010)(0.4771)
C)
0.3010 – 0.4771
D)
0.4471 – 0.3010
E)
0.3010
0.4771
20)
log6
564=
20)
A)
–4
5
B)
–5
4
C)
0
D)
4
5
E)
5
4
21)
If e ln(4x+3) = 7, then x=
21)
A)
–2.
B)
2.
C)
0.
D)
1
4 ln 7
3.
E)
1.
22)
ln 6 + ln 1
6+ ln e2=
22)
A)
0
B)
1
C)
2
D)
3
E)
4
6
23)
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
23)
A)
–ln(0.01)
5.
B)
–
ln 1
5
0.01 .
C)
50e–0.1.
D)
–
1
5
ln(0.01) .
E)
1
5
ln(–0.01) .
24)
If log2(x + 3) = – 2, then x=
24)
A)
–7
B)
–4
3
C)
–11
4
D)
–1
E)
3
4
25)
If 2e3x– 5 = 3, then x=
25)
A)
ln 3
4
B)
3
ln 4
C)
ln 4
3
D)
4
3
E)
1
3ln 7
2
26)
If log 3 = 0.4771, then log 0.3 =
26)
A)
–4.771.
B)
–1.4771.
C)
1.4441.
D)
0.5229.
E)
–0.5229.
27)
If log 2 = 0.3010 and log 3 = 0.4771, then log 18 =
27)
A)
1.9030.
B)
2.3343.
C)
0.0685.
D)
1.2552.
E)
0.1761.
28)
Which of the following is true? If f(x) = ln x, then
28)
A)
both the domain and the range of f are all positive real numbers.
B)
both the domain and the range of f are all real numbers.
C)
the domain of f is all positive real numbers and the range is all real numbers.
D)
the domain of f is all real numbers and the range is all positive real numbers.
E)
the domain of f is all real numbers except zero and the range is all real numbers.
29)
If $1000 is invested for 2 years at 6% compounded quarterly, then the compound amount at the end
of the period is
29)
A)
$1593.84.
B)
$1123.60.
C)
$1120.00.
D)
$1126.49.
E)
$1141.23.
30)
Of the following the best approximation of e is
30)
A)
1.8.
B)
1.4.
C)
3.1.
D)
2.7.
E)
2.3.
31)
log 300
100 =
31)
A)
3
B)
log 200
C)
log 3
D)
2 + log3
2
E)
none of the above
32)
The above graph is best represented by
32)
A)
y=e4
B)
y
=1
4
x
C)
y
= ln 4
D)
y =4x
E)
log4x
33)
If $500 is invested for 3 years at 7% compounded semiannually, then the compound interest at the
end of the period is
33)
A)
$114.63.
B)
$112.52.
C)
$120.37.
D)
$150.36.
E)
$122.56.
C
34)
Writing 2 ln x–1
3ln y+ 4 ln z as a single logarithm gives
34)
A)
ln(x2–
3y+z4).
B)
ln x2z4
y3.
C)
ln 2x–1
3y+ 4z.
D)
ln x2z4
3y
.
E)
ln(x2+z4)
ln 3y
.
35)
log51
25 =
35)
A)
–1
2
B)
1
2
C)
–2
D)
–5
E)
–4
36)
log 10,000 – 6 log 10 =
36)
A)
0
B)
10,000
610
C)
1
D)
–6log 10
E)
4
37)
If log4(x+ 6) = 2 –log4 x, then x=
37)
A)
4.
B)
–4.
C)
10.
D)
2.
E)
8.
38)
If log(x+3)4= 4, then x can equal
38)
A)
44.
B)
12.
C)
–3.
D)
1
10 .
E)
7.
39)
If log 2 = 0.3010 and log 3 = 0.4771, then log 27
2=
39)
A)
0.1761.
B)
3.1761.
C)
1.1303.
D)
04.283.
E)
1.7323.
40)
ln x
yz
=
40)
A)
ln x–1
2(ln y+ ln z)
B)
ln x
ln y ln z
C)
ln x–1
2(ln y– ln z)
D)
ln x+1
2(ln y– ln z)
E)
ln x–ln y +ln z
41)
If $10,000 is invested at 16% compounded quarterly, then the compound amount at the end of six
years is
41)
A)
$24,278.09.
B)
$26,678.42.
C)
$21,173.75.
D)
$26,987.33.
E)
$25,633.04.
42)
Changing log7(4x) to natural logarithms gives
42)
A)
ln
e
ln(4
x
).
B)
ln(4
x
)
ln 7 .
C)
ln(4
x
)
ln
e
.
D)
ln (4
x
).
E)
ln 7
ln(4
x
).
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
43)
A certain medicine reduces the bacteria present by 25% each day. Currently 28,000 bacteria
are present. Make a table of values for the number of bacteria present each day for 0 to 4
days. For each day write an expression for the number of bacteria as a product of 28,000
and a power of 0.75. Use the expressions to make an entry in your table for the number of
bacteria after t days. write a function N for the number of bacteria after t days.
43)
Days Bacteria Expression
028,000 28,000(0.75)0
121,000 28,000(0.75)1
44)
Suppose $2000 is invested at 6.5% compounded quarterly.
(a) Find the value of the investment after 5 years.
(b) Find the value of the interest which was earned over the first 5 years.
44)
45)
Consider h(x) = 2ln(x) – 3
(a) Graph h(x)
(b) Find the domain of h(x)
45)
46)
Graph f(x) =log2x
46)
47)
The population of a city is given by P= 10,000(1.04)t where t is the number of years after
1988. Find the population in
(a) 1988
(b) 1989
(c) 1990.
47)
48)
Find x: log2(x+ 4) = 3
48)
49)
The number of years it takes for an amount which is invested at an annual rate of 7% and
compounded continuously to become m times as large is a function of the enlargement
factor given t(m) =ln(m)
0.07 . Use a graphics calculator to find how many times larger (to the
nearest whole number increment) the investment will be in 10, 20, 30, and 40 years.
49)
50)
Assume that log 5 = 0.6690 and log 6 = 0.7782. Determine the value of log 6
5.
50)
51)
A trust fund is being set up by a single payment so that at the end of 30 years there will be
$20,000 in the fund. If the interest rate is 8% compounded quarterly, how much money
should be paid initially into the trust fund?
51)
52)
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?
52)
53)
If p=320–q, by using common logarithms express q in terms of p.
53)
54)
Solve for x: log x= log 3 + 2 log 4
54)
55)
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.
55)
56)
Find x: logx(6 – 4x–x2) = 2
56)
57)
Write the following in terms of ln x and ln(x+ 1): ln x4
(x+ 1)3
57)
58)
Express 34= 81 in logarithmic form.
58)
59)
Find x and express your answer in terms of common logarithms: 4x= 3
59)
60)
Graph y=f(x) =(0.5)x.
60)
61)
Solve for x: log(98 –x+x2) = 2
61)
62)
Evaluate and simplify: ln e2+ ln 1
62)
63)
Assume that log 6 = 0.7782. Determine the value of log(0.6).
63)
64)
Find x: log x= – 2
64)
65)
Express log21
8= – 3 in exponential form.
65)
66)
Assume log3x= 2; log3y= .12. Find log33xy2.
66)
67)
Write the following in terms of ln x, ln(x– 3), and ln(x+ 1): ln x(x–3)2
x+ 1
67)
68)
An earthquake which is 7200 times as intense as a zero–level earthquake has a magnitude
on the Richter Scale which is 3.6 less than the intensity of another earthquake. What is the
intensity of the other earthquake?
68)
69)
The work done by 1 kilogram sample of nitrogen as its volume changes from an initial
value Vi to a final value of Vf during a constant temperature process is given by W= 3.2×
102ln Vf
Vi. If such a sample expands from a volume of 3 liters to a volume of 5 liters,
determine the work done by the gas.
69)
70)
A trust fund is being set up by a single payment so that at the end of 5 years there will be
$10,000 in the fund. If the interest rate is 3 3
4% compounded quarterly, how much money
should be paid initially into the trust fund?
70)
71)
Given ln x= 7.1; ln y= 8.2, find ln x3
y7.
71)
72)
The demand function for a product is p= 90 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 $18.00?
72)
73)
Solve for x: log3(x+ 7) – 3log3 2 =log3x
73)
74)
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.
74)
75)
Assume that log 5 = 0.6690 and log 6 = 0.7782. Determine the value of log 30.
75)
76)
Write the following in terms of ln(x+ 2) and ln(x+ 4): ln 4(x+ 2)(x+4)3
76)
77)
Given ln x= 7.1; ln y= 8.2, find ln(x3y7)
77)
78)
If $2000 is invested for 3 years at 8% compounded quarterly, find
(a) the compound amount and
(b) the compound interest.
78)
79)
Graph g(x) =1
3
x
79)
x
80)
The population of a city is given by P= 100,000(1.03)t where t is the number of years after
1988. Find the population in
(a) 1988
(b) 1989
(c) 1990.
80)
81)
Your friend started a savings plan with 3 pennies and each day she saved three times the
amount she saved the previous day. Later you started a savings plan with 9 pennies and
each day you saved 9 times the amount you saved the previous day. On the day that your
friend had been saving for 8 less than 4 times as many days as you, you saved the same
amount as your friend. How many days have you been saving?
81)
82)
The demand equation for a product is given by q= 1000 –3p. Solve for p and express your
answer in terms of common logarithms.
82)
83)
Find x: log x= 3
83)
84)
Find x and express your answer in terms of natural logarithms: 2e3x= 6
84)
85)
Evaluate and simplify: ln e3
85)
86)
Simplify: e2 ln x– 3 ln y
86)
87)
Solve for x: log2(x– 4) +log2 3 =log2x
87)
88)
Find x: ln e3=x
88)
89)
The number of yearly visitors to a resort has been shrinking at the rate of 6%. Currently the
resort gets 120,000 tourists each year.
(a) If this rate continues, how many tourists will they get in 15 years?
(b) Use a graphing calculator to predict the number of years until the number of tourists
will be less than 20,000.
89)
90)
Write the following in terms of ln(x+ 2) and ln(x+ 4): ln 1
(x+2)2(x+4)3
90)
20