King Saud University
Preparatory Year Deanship
Math Skill Department
Benchmark : Math 140 Course
Chapter (5)
Exponential and Logarithmic
Functions
Objective (5.1):
Graph Exponential Functions
1) Graph the function by making a table of coordinates.
A)
B)
C)
D)
2
f(x) = 3 x
2) Graph the function by making a table of coordinates.
A)
B)
C)
D)
3
x
f(x) = 3
4
3) Graph the function :
A)
g(x) = 4(x - 4)
B)
C)
D)
4
4) Graph the function. Give the domain and range.
A) domain: (
, 0); range: (
C) domain: (0, ); range: (
f(x) = log
B) domain: (
, )
5
x
, 0); range: (
D) domain: (0, ); range: (
, )
5
, )
, )
5)
Graph the function. Give the domain and range. f(x) = log (x - 4)
2
A) domain: (
C) domain: (0, ); range: (
Objective (5.2):
6)
B) domain: (
, ); range: (-4, )
D) domain: (4, ); range: (
, )
B) 416 = 2
B) 24 = x
C) x2 = 4
8) Write the equation in its equivalent exponential form:
A) 642 = b
log 4 1 6 = 2
C) 24 = 16
7) Write the equation in its equivalent exponential form:
A) 4x = 2
, )
Change from Logarithmic to Exponential Form
Write the equation in its equivalent exponential form.
A) 162 = 4
, ); range: (0, )
B) b2 = 64
log 4 x = 2
log b 64 = 2
C) 2b = 64
6
D) 42 = 16
D) 42 = x
D) 64b = 2
Objective (5.3):
9)
Change From Exponential to Logarithmic Form
Write the equation: 2 3 = x
A) log x = 3
2
B) log x = 2
3
10) Write the equation: 2-2 =
A) log
2
-2 =
1
4
11) Write the equation:
A) logc 1000 = 3
Objective (5.4):
in its equivalent logarithmic form.
1
4
B) log
c3 = 1000
1/2
14)
C) logc 3 = 1000
5
15) Evaluate the expression ( log 5
Objective (5.5):
B)
1
= -2
2 4
D) log3 1000 = c
without using a calculator.
C) 6
D) 8
1 ) without using a calculator.
125
C) 15
D) 1
3
3 ) without using a calculator.
B) 1
1
2
D) log
in its equivalent logarithmic form.
B) -3
1
A)
3
1
=2
-2 4
Evaluate Logarithms
Evaluate the expression ( log 3
A) -
C) log
2 = -2
B) log1000 c = 3
Evaluate the expression ( log
A) 3
D) log 2 = 3
x
in its equivalent logarithmic form.
12) Evaluate the expression ( log2 8 )
A) 2
B) 3
13)
C) log 3 = x
2
C) 3
D)
1
2
D)
1
2
1 ) without using a calculator.
5
1
5
C) -
1
5
Use Basic Logarithmic Properties
16) Evaluate the expression without using a calculator. log 6 1
1
A) 1
B)
C) 0
6
7
D) 6
17) Evaluate the expression without using a calculator.
A) 1
B) 4
C)
1
4
log 4 4
D) 0
18) Evaluate the expression without using a calculator. log (7)18
7
A) log
7
18
Objective (5.6):
B) 25
C) 7
D) 18
Graph Logarithmic Functions
19) The graph of a logarithmic function is given. Select the function for the graph from the
options.
A) f(x) = log x
4
B) f(x) = log (x - 1)
4
C) f(x) = log x - 1
4
D) f(x) = log (x + 1)
4
20) The graph of a logarithmic function is given. Select the function for the graph from the
options.
A) f(x) = log
2
(x + 2)
B) f(x) = log
2
x
C) f(x) = log
8
2
(x - 2)
D) f(x) = log
2
x+2
21) Write an equation for the graph given. The graph represents an logarithmic function f with
base 2 .
A) log2 (x + 4) + 2
B) log2 (x + 4) - 2
C) log2 (x - 4) - 2
D) log2 (x - 4) + 2
22) Write an equation for the graph given. The graph represents an exponential function f with
base 2 .
A) 2x - 1 + 1
Objective (5.7):
B) 2x + 1 + 1
C) 2x + 1 - 1
Find the Domain of a Logarithmic Function
23) Find the domain of the logarithmic function.
A) (-5, )
B) (- , 5) or (5, )
f(x) = log 2 (x - 5)
C) (- , 0) or (0, )
24) Find the domain of the logarithmic function.
A) (- , 0)
D) 2x - 1 - 1
B) (-1, )
h(x) = 1 + log 3 (-x )
C) (0, )
9
D) (5, )
D) (- , )
25)
Find the domain of the logarithmic function.
A) (-™ , 0)
26)
)
Objective (5.8):
f(x) = ln (7 - x)
C) (- , 7) or (7, )
B) (7, )
f(x) = ln
D) (- , 0)
1
x-7
C) (0, )
B) (-7, 8)
D) (-7, )
f(x) = log x + 7
x-8
C) (- , -7)
(8,
D) (- , -7)
)
Use Common Logarithms
B) 5.0592
3 ( log 1 0(5.4) )
C) 16.2
Evaluate or simplify the expression without using a calculator.
A) 30
31)
D) (0, )
Evaluate or simplify the expression without using a calculator.
A) 1.62
30)
B) (-7, )
Find the domain of the logarithmic function.
A) (8,
29)
C) (-1, )
Find the domain of the logarithmic function.
A) (1, )
28)
B) (- , )
Find the domain of the logarithmic function.
A) (- , 7)
27)
(0 , )
1
log (x + 1)
3
B)
10
1
C)
3
Evaluate or simplify the expression without using a calculator.
A) 7
B) log 7
Objective (5.9):
log 1000
D) 3
log (10)7
D) 107
C) 10
32) Evaluate or simplify the expression without using a calculator.
A) 1.56
D) 162
B) 4.9460
3 10
C) 156
D) 15.6
Use Natural Logarithms
33) Evaluate or simplify the expression without using a calculator. ln
A) 4
(log 5.2)
B) 1
4
C) 4e
10
4
e
D) e
4
34) Evaluate or simplify the expression without using a calculator.
A) 153
B) -153
e( ln 153)
D) e153
C) ln 153
35) Use properties of logarithms to write and simplify the logarithmic expression as much as
log 2 (2x)
possible.
A) x
C) 1 + log
B) 2
2
x
D) 1
36) Use properties of logarithms to write and simplify the logarithmic expression as much as
log (1000x)
possible.
A) 3x
B) 30 + log x
Objective (5.11):
C) 3 + log x
Use the Quotient Rule and
D) 3log x
Use the Power Rule
37) Use properties of logarithms to evaluate logarithmic expressions as much as possible.
log
x
100
A) log x - 2
B) 100x
C) -20x
D) log x + 2
38) Use properties of logarithms to evaluate logarithmic expressions as much as possible.
log 7
4
y
A) 1 log y
7
4
B) 4 log y
7
C)
4
1
log 7 y
4
D) 1 log y
4
7
39) Use properties of logarithms to expand the logarithmic expression as much as possible.
Where possible, evaluate logarithmic expressions .
2
log
2 x
A) 1 - log
2
x
B) 2
C)
1
x
D) - log
2
x
40) Use properties of logarithms to expand the logarithmic expression as much as possible.
Where possible, evaluate logarithmic expressions .
e3
ln
5
A) 3 + ln 5
C) ln e3 - ln 5
B) 3 - ln 5
11
D) ln e3 + ln 5
Objective (5.12):
41)
Condense Logarithmic Expressions
Use properties of logarithms to Write the expression as a single logarithm where
possible, evaluate logarithmic expressions.
log 4 (x - 8) - log 4 (x - 4 )
A) log -4
4
B) log x - 8
4 x-4
x-8
C) log
4 x +4
D) log (x2 - 12x + 32)
4
42) Use properties of logarithms to Write the expression as a single logarithm where
possible, evaluate logarithmic expressions.
A)
1
log x - 2
5
2
B) log
5
log 5
x
25
C) 10 -
x-2
1
log x
5
2
D) - 2 log
5
x
43) Use properties of logarithms to Write the expression as a single logarithm where
possible, evaluate logarithmic expressions. 5 ln x A) ln
44)
x5
3
y
B) ln
C) ln x 5 y3
D) ln x 5
3
y
Use properties of logarithms to Write the expression as a single logarithm where
6ln (x - 5) - 11 ln x
possible, evaluate logarithmic expressions.
B) ln
A) ln 66x(x - 5)
Objective (5.13):
45)
x5
y3
1
ln y
3
C) ln
(x - 5)6
x 11
D) ln x 11(x - 5)6
Use Like Bases to Solve Exponential Equations
Solve the equation
A) {4}
46) Solve the equation
A) {1 , -3}
6(x - 5)
11x
4 (1 + 2x) = 6 4
B) {1}
4 (x)
C) {-1}
D) {16}
C) {- 3}
D) {1 , 3}
( 2(x -2) )= 128
B) {3}
12
2(7 - 3x) =
47) Solve the equation
1
2
A)
48)
1
4
B) {1}
e(x + 8) =
Solve the equation
A) {-12}
C) {3}
D) {-3}
C) {4}
D) {-4}
1
e4
B) {12}
49) Solve the exponential equation. Express the solution set in terms of natural logarithms.
e
(x + 4)
=2
A) {ln 2 - 4}
C) {e8 }
B) {ln 6}
D) {e2 + 4}
50) Solve the exponential equation. Express the solution set in terms of natural logarithms.
5 + 3e
(1 - 3x )
=8
1
A) { }
3
B) {3}
C) {- 3}
D) {
-1
}
3
51) Solve the exponential equation. Express the solution set in terms of natural logarithms.
5
(x + 7)
=3
A) {ln 3 - ln 5 - ln 7}
B)
ln 5
+7
ln 3
C)
ln 3
-7
ln 5
D)
ln 5
+ ln 7
ln 3
52) Solv the exponential equation. Express the solution set in terms of natural logarithms.
4
e
A) {
(1 + 3x
- 9e
(5 - 2x)
1
(ln 4 - ln 9) + 5
4
=0
B)
1
9
(4+ ln ( ) )
4
4
C)
13
1
9
( 4 + ln ( ) )
5
4
D)
1
9
(5+ ln ( ) )
5
4
Objective (5.13):
Expand Logarithmic Expressions
53) Use properties of logarithms to expand the logarithmic expression as much as possible.
log b (yz 4 )
Where possible, evaluate logarithmic expressions:
A) log y + 4 log z
b
b
B) 4 log y + 4 log z
b
b
C) log y + log 4z
b
b
D) 4 log yz
b
54) Use properties of logarithms to expand the logarithmic expression as much as possible.
log
Where possible, evaluate logarithmic expressions:
A) log
4
(x - 6) - 5 log
4
C) log (x - 6) - log x
4
4
x
x-6
x5
4
B) 5 log
x - log (x - 6)
4
4
D) log (x - 6) + 5 log x
4
4
55) Use properties of logarithms to expand the logarithmic expression as much as possible.
ln
Where possible, evaluate logarithmic expressions:
A)
1
ln x - ln y
2
Objective (5.14):
B) ln
x - ln
y
C)
x
y
1 x
ln
2 y
D)
1
1
ln x - ln y
2
2
Use the Definition of a Logarithm to Solve Logarithmic Equations
and Use the Logarithms rules to Solve Logarithmic Equations
56) Solve the logarithmic equation.
log (x - 1) = -1
B) - 2
3
C)
A) - 2
57) Solve the logarithmic equation.
A)
4
3
D) 4
log (5x - 5) = log (3x + 7)
6
6
B) 6
58) Solve the logarithmic equation.
A) 5, 20
3
C) 1
D) {2}
log (x - 15) = 2 - log (x)
3
6
B) -™
C) {-5 , 20}
14
D) 20
Objective (5.15):
59) Let
A)
r = ln 2 and s = ln 3 .
s
r
60) Let
A)
B)
r-s
1
5
B)
( s)
C)
s
r
C)
63) Let : log
A) 12
b
A = 3 and
Let : log A = 5 and
b
A) 1
Let :
A) -12
3)
log A = 6 and
b
D)
s-r
D)
s+r
] in terms of ( r ) and ( s) or one of them?
s
D)
1
5
( r)
Express [ ln ( 1 ) ] in terms of ( r ) and ( s) or one of them?
B) -2(r
36
- s)
C) 2(s
- r)
log B = - 4 , Find : log (AB)
b
b
B) 7
C) -12
D) 2(s
D) -1
log B = - 4 , Find : log ( B )
b
b A
C) -1
D) 9
log B = - 8 , Find: log ( A )
b
b
B
B) 14
logb A = 6 and
s-r
C) 5
B) -9
A) 10
66) Let :
5
B) 5r
(s + r)
r+s
Express [ ln (1.5) ] in terms of ( r ) and ( s) or one of them?
r = ln 2 and s = ln 3 .
A) -2
65)
r-s
r = ln 2 and s = ln 3 . Express [ ln (
62) Let
64)
Express [ ln (6) ] in terms of ( r ) and ( s) or one of them?
r = ln 2 and s = ln 3 .
61) Let
A)
Evaluate Logarithms and Use Basic Logarithmic Properties.
C) -2
logb B = - 8 , Find: logb (A2 )
B) 36
C) 3
15
D) -14
D) 12
+ r)
Objective (5.16): Find inverse of logarithmic function
67) Find the inverse of :
f(x) = 3(x -4) + 5
f-1(x) = log 3 (x - 5) + 4
C) f-1(x) = log (x - 5) - 4
3
f-1(x) = log 4 (x - 5) + 4
D) f-1(x) = log (x - 5) + 3
5
A)
68)
Find the inverse of :
B)
f(x) = 2 + log 4 (x - 3)
A) f-1(x) = 4(x+2) + 3
B) f-1(x) = 4(x-2) + 3
C) f-1(x) = 4(x-2) - 3
D) f-1(x) = 4(x+2) - 3
16