Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1) 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
A) 1,040,000.
B) 1,002,000.
C) 1,020,000.
D) 1,004,000.
E) 1,040,400.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
2) A radioactive element is such that N grams remain after t hours, where N = 20e-0.028t.
How many grams remain after 30 hours?
2)
3) 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.
3)
4) Suppose $20,000 is invested at 6.5% compounded annually.
(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.
4)
1
5) Express log2 = -3 in exponential form.
8
5)
6) Evaluate and simplify: log 100
6)
7) Evaluate and simplify: ln e3
7)
8) Evaluate and simplify: log (0.1)
8)
9) Evaluate and simplify: log7 1
9)
10) Evaluate and simplify: ln e
10)
1
81
11)
12) Evaluate and simplify: log2 (-2)
12)
13) Evaluate and simplify: log6 6
13)
14) Find x: log3 x = 3
14)
11) Evaluate and simplify: log3 1
1)
15) Find x: logx = 0
15)
16) Find x: ln x = 1
16)
17) Find x: log4 x = 2
17)
18) Find x: log2 x = -3
18)
19) Find x: log x = 3
19)
20) Find x: ln x = -2
20)
21) Find x: log x = -2
21)
22) Find x: log6 36 = x
22)
1
= x
25
23)
23) Find x: log5 24) Find x: log 100,000 = x
24)
25) Find x: log4 2 = x
25)
26) Find x: log 0.01 = x
26)
27) Find x: ln e3 = x
27)
28) Find x: logx 16 = 4
28)
29) Find x: log2 (x + 4) = 3
29)
30) Find x: logx (4x - 1) = 1
30)
31) Find x: logx (4x -3) = 2
31)
32) Find x: log4 4 6 = x
32)
33) Find x and express your answer in terms of natural logarithms: e4x = 2
33)
34) Find x and express your answer in terms of natural logarithms: 2e3x = 6
34)
35) 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.
35)
2
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
36) If 2e3x - 5 = 3, then x =
1 7
ln 4
A) ln B)
3 2
3
36)
3
C)
ln 4
ln 3
D)
4
4
E)
3
37) 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
0.03
0.03
ln 2
ln 2
ln 2
.
B)
.
C) - .
D)
.
E) - .
A)
ln 2
ln 2
0.03
0.03
40
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
38) Find x: logx (6 - 4x - x 2 ) = 2
38)
39) Express 2(ln 4 + ln 3 - 3 ln 2) as a single logarithm.
39)
40) Express 2 log(x) - 3 log(x + 7) as a single logarithm.
40)
41) Solve for x: 3 log3 x+log3 4 = 8
41)
42) Find x and express your answer in terms of common logarithms: 4 x = 3
42)
43) Find x and express your answer in terms of natural logarithms: 2 -x - 3 = 8
43)
44) Find x and express your answer in terms of common logarithms: 102x-3 = 4
44)
45) Solve for x: log x = log 3 + 2 log 4
45)
46) Solve for x: ln x + ln 3 = ln(x + 1)
46)
47) Solve for x: log(x + 1) - log(x - 2) = 1
47)
48) Solve for x: 2 log2 x + log2 5 = 7
48)
3
37)
Answer Key
Testname: MATH155EXPONENT_LOG
1) E
2) 8.634 gms
Days
Bacteria
0
28,000
1
21,000
2
15,750
3
11,812.5
4
8859.4
3)
t
Expression
28,000(0.75)0
28,000(0.75)1
28,000(0.75)2
28,000(0.75)3
28,000(0.75)4
28,000(0.75)t
Equation N(t) = 28,000(0.75)t
4) (a) $27,401.73
(b) $7,401.73
1
5) 2 -3 = 8
6)
7)
8)
9)
2
3
-1
0
1
10)
2
11) -4
12) not defined
13) 1
14) 27
15) 1
16) e
17) 16
1
18)
8
19) 1000
20) e-2
21)
1
100
22) 2
23) -2
24) 5
1
25)
2
26) -2
27) 3
28) 2
29) 4
1
30)
3
31) 1, 3
32) 6
4
Answer Key
Testname: MATH155EXPONENT_LOG
33)
ln 2
4
34)
ln 3
3
35)
ln 2 0.69315
≈ ≈ 17.33 days
0.04
0.04
36) B
37) D
38) 1
39) ln 9
4
40) log
x2
(x + 7)3
41) 2
log3
42)
log4
43) - 44)
ln 11
ln 2
3 + log 4
2
45) 48
1
46)
2
47)
7
3
7
48) x = = 1.4
5
5