Exam
Name___________________________________
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
1) The function D(h) = 5e-0.4h can be used to determine the milligrams D of a certain drug in
a patient's bloodstream h hours after the drug has been given. How many milligrams (to
two decimals) will be present after 7 hours?
1)
2) The formula P = 14.7e-0.21x gives the average atmospheric pressure, P, in pounds per
square inch, at an altitude x, in miles above sea level. Find the average atmospheric
pressure for an altitude of 2.3 miles. Round your answer to the nearest tenth.
2)
3) The function f(x) = 500(0.5)x/60 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. Find the amount of radioactive material in the vault after 160 years.
Round to the nearest whole number.
3)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
The graph of an exponential function is given. Match the graph to one of the following functions.
4)
y
10
5
-10
-5
5
10
x
-5
-10
A) f(x) = 4 x
B) f(x) = 4 x - 2
C) f(x) = 4 x - 2
1
D) f(x) = 4 x + 2
4)
5)
5)
y
10
5
-10
-5
5
10
x
-5
-10
A) f(x) = 2 x
B) f(x) = 2 x - 1
C) f(x) = 2 x + 1
D) f(x) = 2 x + 1
6)
6)
y
10
5
-10
-5
5
10
x
-5
-10
A) f(x) = -4 x
B) f(x) = 4 x
C) f(x) = -4 -x
D) f(x) = 4 -x
7)
7)
y
10
5
-10
-5
5
10
x
-5
-10
A) f(x) = - 2 -x
B) f(x) = - 2 x
C) f(x) = 2 -x
2
D) f(x) = 2 x
8)
8)
y
10
5
-10
-5
5
10
x
-5
-10
B) f(x) = - 5 x
A) f(x) = - 5 -x
C) f(x) = 5 -x
D) f(x) = 5 x
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Graph the function.
9) f(x) = -1 + ex
9)
y
10
5
-10
-5
5
10
x
-5
-10
10) f(x) = ex
10)
y
6
4
2
-6
-4
-2
2
4
6
x
-2
-4
-6
3
11) f(x) = e3x - 2
11)
y
5
4
3
2
1
-5 -4 -3 -2 -1
-1
1
2
3
4
5
x
-2
-3
-4
-5
Change the exponential expression to an equivalent expression involving a logarithm.
12) 7 3 = 343
13) 3 -3 =
1
27
12)
13)
14) 4 3 = x
14)
15) 4 3/2 = 8
15)
16) 10x = 1000
16)
17) ex = 10
17)
Convert to a logarithmic equation.
18) 2 3 = 8
18)
19) e-3 = 0.04979
19)
20) e-4 = t
20)
21) yz = 9
21)
1
4
22)
23) 6 2 = 36
23)
22) 2 -2 =
Change the logarithmic expression to an equivalent expression involving an exponent.
24) log 1/4 16 = -2
4
24)
1
25) log 3 = -2
9
25)
26) log 4 16 = 2
26)
27) log 2 x = 3
27)
28) log b 25 = 2
28)
29) log 7 49 = x
29)
2
3
30)
30) logb 49 =
31) ln x = 6
32) ln
31)
1
= -4
e4
32)
Convert to an exponential equation.
33) log6 216 = 3
33)
34) log6 36 = t
34)
35) log6 1 = 0
35)
Find the exact value of the logarithmic expression.
36) log5 125
36)
37) log5
1
25
37)
38) log7
1
343
38)
39) log4
1
64
39)
40) log 8 1
40)
41) log1/4 256
41)
5
42) log 9
9
42)
43) log 10 10,000
43)
44) ln l
44)
45) ln e
45)
46) ln e7
46)
Solve the equation.
47) log 5 x = 2
47)
48) log 2 8 = x
48)
49) log8 x2 = 4
49)
50) log 2 (x + 1) = 3
50)
51) log 6 (x + 2) = -1
51)
52) log5 (x2 - 4x) = 1
52)
53) log42 (x2 - x) = 1
53)
54) 9 ln 5x = 36
54)
55) 3 + 8 ln x = 10
55)
56) ln
56)
x+2=2
57) e2x = 7
58) e
x+6
57)
=8
58)
Write as the sum and/or difference of logarithms. Express powers as factors.
14 x
59) log 5
y
60) log 5
x5
y6
59)
60)
6
61) log 4
x+6
x3
61)
62) log w
11x
4
62)
63) log 3
6x
63)
x
64) log 4
16
64)
8
65) log 9
17
2
s r
65)
66) log 6
pq
5
66)
67) ln
(x + 1)(x - 9) 3/2
,
(x - 2)3
x>9
67)
Express as a single logarithm.
68) log c x + log c y
68)
69) 6 log b q - log b r
69)
70) 10 log a 2 + 7 log a 7
70)
71) ( log a t - log a s) + 5 log a u
71)
72) 6 log a m -
3
1
log a n + log a j - 5 log a k
5
4
72)
73) 3 log 6 x + 5 log 6 (x - 6)
73)
74) 3 loga (2x + 1) - 2 loga (2x - 1) + 2
74)
75) How long will it take for $2900 to grow to $25,000 at an interest rate of 4.5% if the interest is
compounded continuously? Round the number of years to the nearest hundredth.
75)
76) How long will it take for $7000 to grow to $25,200 at an interest rate of 10% if the interest is
compounded quarterly? Round the number of years to the nearest hundredth.
76)
Solve.
7
77) Suppose that $12,000 is invested at an interest rate of 5.8% per year, compounded
continuously. What is the doubling time?
77)
78) Suppose that $9000 is invested at an interest rate of 5.2% per year, compounded
continuously. What is the balance after 6 years?
78)
79) Randy invested his inheritance in an account that paid 6.5% interest, compounded
continuously. After 5 years, he found that he now had $49,368.37. What was the original
amount of his inheritance?
79)
80) Kimberly invested $6000 in her savings account for 8 years. When she withdrew it, she
had $7097.62. Interest was compounded continuously. What was the interest rate on the
account?
80)
81) How long will it take for the population of a certain country to double if its annual growth
rate is 2.7%? Round to the nearest year.
81)
82) If the population of a certain country doubles in 11.00 years, find the growth rate k.
Assume that the population increases exponentially.
82)
83) The number of books in a small library increases according to the function B = 6400e0.04t,
where t is measured in years. How many books will the library have after 4 year(s)?
83)
84) The population of a small country increases according to the function B = 2,400,000e0.04t,
where t is measured in years. How many people will the country have after 8 years?
84)
85) How long will it take for prices in the economy to double at a 7% annual inflation rate?
(Round to the nearest year.)
85)
86) How long will it take for the population of a certain country to triple if its annual growth
rate is 3.8%? Round to the nearest year.
86)
Solve the problem.
87) A sample of 800 grams of radioactive substance decays according to the function
A(t) = 800e-0.04t, where t is the time in years. How much of the substance will be left in the
87)
sample after 20 years? Round to the nearest whole gram.
88) How long will it take a sample of radioactive substance to decay to half of its original
amount, if it decays according to the function A(t) = 750e-0.144t, where t is the time in
88)
years? Round to the nearest hundredth year.
89) A certain radioactive isotope has a half-life of approximately 1700 years. How many years
would be required for a given amount of this isotope to decay to 75 % of that amount?
89)
90) An artifact is discovered at a certain site. If it has 56 % of the carbon-14 it originally
contained, what is the approximate age of the artifact? (carbon-14 decays at the rate of
0.012% annually.)
90)
8
91) A certain radioactive isotope decays at a rate of 0.275 % annually. Determine the half-life
of this isotope, to the nearest year.
91)
92) A certain radioactive isotope has a half-life of 277 years. Determine the annual decay rate,
k.
92)
93) There are currently 77 million cars in a certain country, decreasing exponentially by 5.5 %
annually. How many years will it take for this country to have 67 million cars? Round to
the nearest year.
93)
9
Answer Key
Testname: TEST3_SAMPLE_PAPER_MTH163
1) 0.3 mg
2)
3)
4)
5)
6)
7)
8)
9)
9.1 lb/in.2
79 pounds
A
D
A
C
A
y
10
5
-10
-5
5
10
x
6
x
-5
-10
10)
y
6
4
2
-6
-4
-2
2
4
-2
-4
-6
11)
y
5
4
3
2
1
-5 -4 -3 -2 -1
-1
1
2
3
4
5
x
-2
-3
-4
-5
12) log 7 343 = 3
10
Answer Key
Testname: TEST3_SAMPLE_PAPER_MTH163
1
13) log 3
= -3
27
14) log 4 x = 3
3
15) log 4 8 =
2
16) log 10 1000 = x
17) ln 10 = x
18) 3 = log 2 8
19) -3 = log e 0.04979
20) ln t = -4
21) z = log y 9
1
22) -2 = log 2
4
23) 2 = log 6 36
24)
1 -2
= 16
4
25) 3 -2 =
1
9
26) 4 2 = 16
27) 2 3 = x
28) b2 = 25
29) 7 x = 49
30) b2/3= 49
31) e6 = x
1
32) e-4 =
e4
33) 6 3 = 216
34) 6 t = 36
35) 6 0 = 1
36) 3
37) -2
38) -3
39) -3
40) 0
41) -4
1
42)
2
43) 4
44) 0
45) 1
46) 7
47) {25}
48) {3}
11
Answer Key
Testname: TEST3_SAMPLE_PAPER_MTH163
49) {64, -64}
50) {7}
11
51) 6
52) {5, -1}
53) {-6, 7}
e4
54)
5
7/8
55) {e
}
4
56) {e - 2}
57)
ln 7
2
58) {ln 8 - 6}
1
59) log 5 14 +
log 5 x - log 5 y
2
60) 5 log 5 x - 6 log 5 y
61) log 4 (x + 6) - 3 log 4 x
62) log w 11 + log w x - log w 4
63)
1
1
log 3 6 + log 3 x
2
2
64)
1
log 4 x - 2
2
65)
1
log 9 17 - 2 log 9 s - log 9 r
8
66)
1
1
1
log 6 p + log 6 q - log 6 5
2
2
2
67)
3
3
9
ln (x + 1) + ln (x - 9) - ln (x - 2)
2
2
2
68) log c xy
q6
69) log b
r
70) log a 2 107 7
tu5
71) log a
s
m 6 j1/4
72) log a
n 3/5 k5
5
73) log 6 x3 (x - 6)
a 2 (2x + 1)3
74) loga
(2x - 1)2
75) 47.87 yr
76) 12.97 yr
77) 12 yr
12
Answer Key
Testname: TEST3_SAMPLE_PAPER_MTH163
78) $12,295.39
79) $35,670
80) 2.1%
81) 26 yr
82) 6.3%
83) 7510
84) 3,305,107
85) 10 yr
86) 29 yr
87) 359 g
88) 4.81 yr
89) 706 yr
90) 4832 yr
91) 252 yr
92) 0.25%
93) 3 yr
13