MTH 112 Practice Test 3
Sections 3.3, 3.4, 3.5, 1.9, 7.4, 7.5, 8.1, 8.2
Use properties of logarithms to expand the logarithmic
expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
x-6
1) log
4 x5
2) log
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.
13) log (x - 1) = -1
3
14) log (x + 3) + log (x - 3) = 2
4
4
x2
2 y7 w4
15) ln 6 + ln (x - 1) = 0
16) log 3 (x + 6) + log 3 (x - 6) - log 3 x = 2
3) log
4
7x
17) log (x + 6) = 3 - log x
6
6
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.
4) 3ln a - 9 ln b - ln c
5) 5 log3 2 +
18) log (x + 20) - log 2 = log (3x + 4)
19) ln (x - 6) + ln (x + 1) = ln (x - 15)
Solve the problem.
20) Find out how long it takes a $2600 investment
to double if it is invested at 8% compounded
semiannually. Round to the nearest tenth of a
r nt
year. Use the formula A = P 1 +
.
n
1
1
log3 (r - 2) - log3 r
5
2
Use common logarithms or natural logarithms and a
calculator to evaluate to four decimal places
6) log 17
π
Solve.
7) log
26
382
21) 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 = 3100e0.046t. When will the account
Solve the equation by expressing each side as a power of
the same base and then equating exponents.
(1 + 2x)
8) 4
= 64
be worth $3902 ?
9) 2 (7 - 3x) =
1
4
22) The function A = A0 e-0.00693 x 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 900 pounds of the material are
initially put into the vault, how many pounds
will be left after 30 years?
Solve the exponential equation. Express the solution set in
terms of natural logarithms.
10) 5 x + 7 = 3
11) 4 x + 4 = 5 2x + 5
12) e x + 4 = 2
1
23) The population of a particular country was 22
million in 1984; in 1994, it was 31 million. The
exponential growth function A =22ekt
Graph the equation.
33) (x - 6)2 + (y - 3)2 = 9
describes the population of this country t years
after 1984. Use the fact that 10 years after 1984
the population increased by 9 million to find k
to three decimal places.
y
10
5
24) The function A = A0 e-0.0077 x models the
-10
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 700 pounds of the material are
placed in the vault, how much time will need
to pass for only 150 pounds to remain?
-5
5
10
x
-5
-10
Graph the equation and state its domain and range. Use
interval notation
34) x2 + y2 = 49
25) The half-life of silicon-32 is 710 years. If 50
grams is present now, how much will be
present in 200 years? (Round your answer to
three decimal places.)
y
10
26) The population of a certain country is growing
at a rate of 1.8% per year. How long will it take
for this country's population to double? Use
ln 2
the formula t =
, which gives the time, t,
k
5
-10
-5
for a population with growth rate k, to double.
(Round to the nearest whole year.)
5
10
x
-5
-10
Write the standard form of the equation of the circle with
the given center and radius.
27) (-10, 0); 3
Solve the system by the substitution method.
35) 8x - y = 1
y = x2 + 6
28) (8, -3); 9
36) x2 + y2 = 113
x + y = 15
Find the center and the radius of the circle.
29) (x + 8)2 + (y - 1)2 = 4
37) xy = 56
x + y = -15
Complete the square and write the equation in standard
form. Then give the center and radius of the circle.
30) x2 + y2 - 4x - 10y + 29 = 36
Solve the system by the addition method.
38) 5x2 - 5y2 = -35
31) 10x2 + 10y2 = 100
2x2 + 2y2 = 50
32) x2 + y2 - 10x - 8y + 29 = 0
Solve the system by the substitution method.
39) y = x2 - 3
x2 + y2 = 5
2
Graph the inequality.
40) (x - 1)2 + (y - 5)2 > 9
43) y > x2
10x + 6y ≤ 60
y
-10
y
10
10
5
5
-5
5
10
x
-10
-5
-5
-5
-10
-10
41) y ≤ x2 - 5
5
10
x
5
10
x
5
10
x
44) x2 + y2 ≤ 36
x2 + y2 ≥ 25
y
10
y
10
5
5
-10
-5
5
10
x
-10
-5
-5
-5
-10
-10
Graph the solution set of the system of inequalities or
indicate that the system has no solution.
42) x + 2y ≥ 2
x-y≤0
45) x2 + y2 ≤ 36
-8x + 3y ≤ -24
y
y
10
10
5
5
-10
-5
5
10
-10
x
-5
-5
-5
-10
-10
3
46) x2 + y2 ≤ 49
y - x2 > 0
Solve the system of equations using matrices. Use
Gaussian elimination with back-substitution.
52) 6x - y - 6z = -43
-6x - 4z = -38
9y + z = 17
y
10
5
-10
-5
53)
5
10
x
-5
Solve the system of equations using matrices. Use
Gauss-Jordan elimination.
54)
3x - y - 7z = 7
6x + 4y - 3z = 67
-6x - 3y + z = -62
-10
Write the augmented matrix for the system of equations.
47)
5x + 5z = 40
5y + 5z = 10
9x + 8y + 7z = 58
Use Gaussian elimination to find the complete solution to
the system of equations, or state that none exists.
55) x + y + z = 7
x - y + 2z = 7
2x + 3z = 14
Write the system of linear equations represented by the
augmented matrix. Use x, y, z, and, if necessary, w for the
variables.
48)
5 9 8 -2
4 0 5 4
7 5 0 2
Write the system of linear equations represented by the
augmented matrix. Use x, y, z, and, if necessary, w for the
variables. Then use back-substitution to find the solution.
49)
1 2 9 -8
0 1 -3 4
0 0 1 -6
56)
x + 8y + 8z = 8
7x + 7y + z = 1
8x + 15y + 9z = -9
57)
4x - y + 3z = 12
x + 4y + 6z = -32
5x + 3y + 9z = 20
58)
Perform the matrix row operation (or operations) and
write the new matrix.
50)
4 -5 1 1
-4 0 5 -2 -4R 1 + R2
-1 3 -3 -1
51)
33 12 -3 -15
1 13 -3
0
2 -7 4 21
x - y + 5z = 11
2x + z = 3
x + 2y + z = 11
1
R
3 1
4
x+ y+z = 9
2x - 3y + 4z = 7
x - 4y + 3z = -2
Answer Key
Testname: MTH 112 PRACTICETEST3
1) log (x - 6) - 5 log x
4
4
2) 2 log x - 7 log y - 4 log 2 w
2
2
1
1
3) log 7 + log x
4
4
2
2
4) ln
33)
5
a3
b9 c
5) log3
32
-10
5
-5
r-2
r
10 x
5
-5
6)
7)
8)
9)
2.4750
1.8248
{1}
{3}
ln 3
10)
-7
ln 5
11)
y
10
-10
Domain = (3, 9), Range = (0, 6)
34)
y
10
5 ln 5 - 4 ln 4
ln 4 - 2 ln 5
5
12) {ln 2 - 4}
4
13)
3
-10
-5
10 x
5
-5
14) {5}
7
15) { }
6
-10
Domain = (-7, 7); Range = (-7, 7)
35) {(7, 55), (1, 7)}
36) {(8, 7), (7, 8)}
37) {(-7, -8), (-8, -7)}
38) {(3, 4), (-3, 4), (3, -4), (-3, -4)}
39) {(-2, 1), (-1, -2), (1, -2), (2, 1)}
16) {12}
17) {12}
12
18)
5
19) ∅
20) 8.8 years
21) 2005
22) 731 pounds
23) 0.034
24) 200 years
25) 41.131
26) 39 years
27) (x + 10)2 + y2 = 9
y
10
5
-10
28) (x - 8)2 + (y + 3)2 = 81
29) (-8, 1), r = 2
30) (x - 2)2 + (y - 5)2 = 36
-5
5
-5
-10
40)
(2, 5), r = 6
31) x2 + y2 = 10
(0, 0), r = 10
32) (x - 5)2 +(y - 4)2 = 12
(5, 4), r = 2 3
5
10
x
Answer Key
Testname: MTH 112 PRACTICETEST3
41)
44)
y
y
10
6
4
5
2
-10
-5
10 x
5
-8
-6
-4
-2
2
4
6
8x
-2
-5
-4
-6
-10
42)
45)
y
y
6
10
4
5
2
-6
-4
-2
2
4
6
-10
x
-5
5
10
x
5
10
x
-5
-2
-4
-10
-6
y
10
43)
y
5
10
5
-10
-5
-5
-10
-5
5
10
x
-10
-5
46)
47)
-10
5 0 5 40
0 5 5 10
9 8 7 58
48) 5x + 9y + 8z = -2
4x + 5z = 4
7x + 5y = 2
49) {(74, -14, -6)}
50)
4 -5 1
1
20
1
-20
-6
-1 3 -3 -1
6
Answer Key
Testname: MTH 112 PRACTICETEST3
51)
11 4 -1 -5
1 13 -3 0
2 -7 4 21
52) {(1, 1, 8)}
53) {(0, 4, 3)}
54) {(7, 7, 1)}
3z
z
55) {(+ 7, , z)}
2
2
56) ∅
57) ∅
58) {(-
7z 34 2z 11
,
, z)}
+
+
5
5 5
5
7