Free Printable Change of Base, Graphs of Exponentials/Logs, Properties of Logs

1.

Condense the Logarithm $5\log_{ }a\ -\ 25\log_{ }b$

a)

$\log\ \left(a^5+b^{25}\right)$

b)

$\log\ \left(a^5-b^{25}\right)$

c)

$\log\ \left(ab\right)^{25}$

d)

$\log_{ }\ \left(\frac{a^5}{b^{25}}\right)$

2.

Condense $3\log_{ }x+4\log_{ }y\ +\log_{ }z$

a)

$\log_{ }\ x^3y^4z$

b)

$12\log_{ }\ xyz$

c)

$\log_{ }\ 3x4yz$

d)

logx³y³z³

3.

Condense: $\log_4\left(x+4\right)-\log_4\left(x-5\right)$

a)

$\log_49$

b)

$\log_4\left(2x-1\right)$

c)

$\log_4\left(x+\text{4}\right)\left(x-5\right)$

d)

$\log_4\left(\frac{x+4}{x-5}\right)$

4.

Condense: $2\log_3\left(11x\right)$

a)

$\log_3\left(22x\right)$

b)

$\log_3\left(121x\right)$

c)

$\log_3\left(11x\right)^2$

d)

$\log_3\left(11x^2\right)$

5.

Expand using the properties of Logaritms $\log_{ }\ \frac{x^3}{y^4z}$

a)

$\log_{ }x+4\log_{ }y\ +\log_{ }z$

b)

$3\log_{ }x-4\log_{ }y\ -\log_{ }z$

c)

$3\log_{ }x+4\log_{ }y\ +\log_{ }z$

d)

$3\log_{ }x-4\log_{ }y\ +\log_{ }z$

6.

Expand the logarithm.
$\log\frac{x}{y^6}$

a)

$\log x+6\log y$

b)

$\log x-6\log y$

c)

$\log x+\log6y$

d)

$\log x-\log6y$

7.

Expand . $\log_6\left(\frac{5x^3}{y}\right)$

a)

$\log_65x^3-\log_6y$

b)

$\log_65+\log_6x^3-\log_6y$

c)

$\log_65+3\log_6x-\log_6y$

d)

$\log_65+3\log_6x+\log_6y$

8.

What is the correct transformed equation for the graph?

a)

y = log(x - 4)

b)

y = log(x) - 4

c)

y = log(x + 4)

d)

y = log(x) + 4

9.

What is the equation of the asymptote on the graph of $y=\log_5\left(x+3\right)+5$ ?

a)

x = -3

b)

x=5

c)

y= -3

d)

y=5

10.

Which of the following has a vertical asymptote at $x=4$

a)

$y=2^x-4$

b)

$y=\log_2\left(x-4\right)$

c)

$y=2^x+4$

d)

$y=\log_2\left(x+4\right)$

11.

What is the transformation of the logarithmic function
$f\left(x\right)=\log\ \left(x+3\right)-1$

a)

left 3, right 1

b)

right 3, down 1

c)

right 3, up 1

d)

left 3, down 1

12.

Which graph matches the function?

a)

b)

c)

d)

13.

What is the range?

a)

(-∞, ∞)

b)

(0, ∞)

c)

[1, ∞)

d)

(1, ∞)

14.

What is the domain?

a)

(-∞, ∞)

b)

(0, ∞)

c)

[1, ∞)

d)

(1, ∞)

15.

Logarithmic functions are the inverse of...

a)

Linear Functions

b)

Exponential Functions

c)

Quadratic Functions

d)

Polynomial Functions

16.

Which is the product property of logarithms?

a)

$\log_b\left(x\cdot y\right)\ =\log_bx\ +\log_by$

b)

$\log_b\left(\frac{x}{y}\right)\ =\ \log_bx\ -\ \log_by$

c)

$\log_bx^n=n\log_bx$

17.

What is the quotient property of logarithms?

a)

$\log_bx^n=n\log_bx$

b)

$\log_b\left(\frac{x}{y}\right)=\log_bx\ -\log_by$

c)

$\log_b\ \left(x\cdot y\right)=\log_bx\ +\ \log_by$

18.

Which is the power property of logarithms?

a)

$\log_b\left(x\cdot y\right)=\log_bx\ +\log_by$

b)

$\log_b\left(\frac{x}{y}\right)\ =\ \log_bx\ -\log_by$

c)

$\log_b\ x^n\ =n\log_bx$

19.

Expand: $\log_b\ \frac{m}{n}$

a)

$\log_bm\ -\log_bn$

b)

$\log_bm\ +\ \log_b\ n$

c)

$m\log_bn$

d)

$n\log_bn$

20.

Expand: $\log\ \frac{k^3}{2}$

a)

$3\log2\ -\ \log k$

b)

$3\log k-\log2$

c)

$3\log k+\log2$

d)

$\log k-3\log2$

21.

Expand: $\ln\ x^5y^2$

a)

5lnx + ln2y

b)

lnx5y2

c)

5lnx + 2lny

d)

5lnx - 2lny

22.

Condense: $\log_25+\log_2x$

a)

$\log_2x^5$

b)

$\log_2\ \frac{5}{x}$

c)

$\log_2\ \frac{x}{5}$

d)

$\log_25x$

23.

Condense: $\log_65-\log_62$

a)

$\log_610$

b)

$\log_6\ \frac{5}{2}$

c)

$\log_6\ 5^2$

d)

$\log_6\ \frac{2}{5}$

24.

Condense: $3\log_4x$

a)

$\log_4\ 3x$

b)

$\log_4x^3$

c)

$\log_4\ \frac{x}{3}$

d)

$\log_4\ 3^x$

25.

Condense: $4\log_5k-\log_5d$

a)

$\log_5k^4d$

b)

$\log_5\ 4kd$

c)

$\log_5\ \frac{k^4}{d}$

d)

$\log_5\ \frac{k}{d^4}$

26.

Condense: $4\log p+2\log q-3\log r$

a)

$\log\ p^4q^2r^3$

b)

$\log\ \frac{p^4}{q^2r^3}$

c)

$\log\ \frac{8pq}{3r}$

d)

$\log\ \frac{p^4q^2}{r^3}$

27.

Condense: $3\ln x\ +\ 8\ln y$

a)

$\ln\left(\frac{x^3}{y^8}\right)$

b)

$\ln\left(x^3y^8\right)$

c)

$24\ln\left(xy\right)$

d)

$11\ln\left(xy\right)$

28.

Condense the logarithmic expression.

a)

A

b)

B

c)

C

d)

D

29.

$\frac{1}{2}\log_{4\ }9$

Condense into a single logarithm. Simplify if possible.

a)

$\log18$

b)

$\log_4\ \frac{9}{2}$

c)

$\log_44.5$

d)

$\log_43$

30.

Condense this expression to a single logarithm.

a)

b)

c)

d)

31.

$\log_312$
Which of the following is a possible expansion for this logarithm? Check all that apply.

a)

$\log_34+\log_33$

b)

$\log_310+\log_32$

c)

$\log_32+\log_36$

d)

$\log_336-\log_33$

e)

$\log_315-\log_33$

32.

Use the properties of logarithms to rewrite as the difference of two logs:

\log_545

a)

$\log_590\ -\log_52$

b)

$\log_515-\log_53$

c)

$\frac{\log45}{\log\ 5}$

d)

$\log_550-\log_55$

33.

Rewrite as a single logarithm:

\log3\ +\ \log7

a)

log 10

b)

log 21

c)

log 3/7

d)

log 3/log 7

34.

Rewrite as a single logarithm: $\log_65-\log_62$

a)

$\log_610$

b)

$\log_6\ \frac{5}{2}$

c)

$\log_6\ 5^2$

d)

$\log_6\ \frac{2}{5}$

35.

Rewrite as a single logarithm:

\log_260\ -\ \log_210

a)

$\log_26$

b)

$\log_250$

c)

$\frac{\log_260}{\log_210}$

d)

$\log_270$

36.

Condense: $3\log_4x$

a)

$\log_4\ 3x$

b)

$\log_4x^3$

c)

$\log_4\ \frac{x}{3}$

d)

$\log_4\ 3^x$

37.

Expand: $\log\left(\frac{3}{2}\right)$

a)

$\log3+\log2$

b)

$3\log2$

c)

$\log(3-2)$

d)

$\log3-\log2$

38.

Use the properties of logarithms to rewrite as the sum of two logarithms:

\log55

a)

log 40 + log 15

b)

log 50 + log 5

c)

log 11 + log 5

39.

Which of the logarithms below is equivalent to the following:
$2\log12$

a)

log 10

b)

log 6

c)

log 24

d)

log 144

40.

Simplify: log₄32+log₄2

(a)

41.

Simplify:

log₃54 - log₃2

(a)

42.

Simplify: log₁₃13⁶

(a)

43.

Simplify 10^log47

(a)

44.

Use these and other properties of logarithms to evaluate the expression.

$\log_232\ -\ 6^{\log_63}$

(a)

45.

Expand

a)

A

b)

B

c)

C

d)

D

46.

log(4x)

a)

log4-logx

b)

log4+logx

c)

4logx

d)

xlog4

47.

log(xy²)

a)

logx+2logy

b)

logx+logy²

c)

logx-2logy

d)

logx+logy+log2

48.

$\log\left(\frac{8x}{yz}\right)\$

a)

log8+logx-logy-logz

b)

log8+logx-logy+logz

c)

log(8x)-log(yz)

d)

8logx-ylogz

49.

Which equation is the inverse of the equation above?

a)

b)

c)

d)

50.

Logarithmic functions have...

a)

Vertical asymptote at x=0

b)

Vertical asymptote at x=1

c)

Horizontal asymptote at y=1

d)

Horizontal asymptote at y=0

51.

Find the corresponding graph of
$\log_3\left(x-1\right)+4$

a)

b)

c)

d)

52.

Evaluate log_b(b)

a)

-1

b)

0

c)

1

d)

b

53.

a)

log₅(x²z²y¹⁰)

b)

log₅(z²y²x¹⁰)

c)

log(zy²x¹⁰)

d)

log₅(z² + y²+ x¹⁰)

54.

Expand completely: log(2x⁵)

a)

log 2 + 5log x

b)

5log 2 + 5log x

c)

5log 2 + log x

d)

log 10 + log x

55.

Which graph matches the equation

a)

b)

c)

d)

56.

Change $\log_{12}⁡2$ into the form of $\frac{\log⁡a}{\log b}$ , where $a$ and $b$ are integers

a)

$\frac{\log12}{\log2}$

b)

$\frac{\log⁡2}{\log12}$

57.

Use the Change of Base Formula to evaluate the expression.

\log_927

a)

$\frac{3}{2}$

b)

$\frac{2}{3}$

c)

$\frac{1}{3}$

d)

3

58.

Solve the log by using change of base

log₄32

a)

8

b)

5/2

c)

2/5

d)

3/2

59.

Simplify $2\cdot\ln e^5$

(a)

60.

Use the change of base rule to expand this problem:

$\log\left(35\right)$

a)

$\frac{\log\left(10\right)}{\ln\left(35\right)}$

b)

$\frac{\log\left(35\right)}{\ln\left(10\right)}$

c)

$\frac{\log_3\left(35\right)}{\log_3\left(10\right)}$

d)

$\frac{\log_9\left(10\right)}{\log_9\left(35\right)}$

61.

$Evaluate\ \log_5\sqrt{5}$

(a)

62.

$Evaluate\ \log_7\frac{1}{49}$

(a)

63.

Evaluate the following logarithm:

log₈₁27

a)

²/₃

b)

³/₂

c)

³/₄

d)

⁴/₃

64.

Evaluate the following logarithm:

log₆6

(a)

65.

Evaluate the following logarithm:

log₂₅125

a)

²/₃

b)

³/₂

c)

³/₄

d)

¹/₂

66.

Evaluate the following logarithm:

log₈4

a)

²/₃

b)

³/₂

c)

³/₄

d)

¹/₂

67.

What are the transformations of this graph?

a)

Right 1 Up 5

b)

Right 1 Down 5

c)

Left 1 Up 5

d)

Left 1 Down 5

68.

Match the graph with its equation

a)

y = log₄ (x)+2

b)

y = log₄ (x + 2) + 1

c)

y = log₄ (x - 1) + 2

d)

y = log₄ (-x + 2)

69.

Determine the range for the function.
$f\left(x\right)=2^{x-4}+5$

a)

y > 4

b)

y > 2

c)

y > 3

d)

y > 5

70.

Write the equation of the function f(x)=log₃x after the following transformations:

Translate 6 units left and 4 units up.

a)

g(x)=log₃(x+6)+4

b)

g(x)=log₃(x+4)-6

c)

g(x)=log₃(x-6)+4

d)

g(x)=3log(x+6)+4

71.

a)

A

b)

B

c)

C

d)

D

72.

a)

A

b)

B

c)

C

d)

D

73.

a)

A

b)

B

c)

C

d)

D

74.

a)

A

b)

B

c)

C

d)

D

75.

a)

A

b)

B

c)

C

d)

D

76.

a)

A

b)

B

c)

C

d)

D

77.

Solve $\log_2\left(x+8\right)=\log_264$

a)

x=16

b)

x=24

c)

x=56

d)

x=40

78.

Solve $\log_8\left(3x+7\right)=\log_8\left(7x+4\right)$

a)

$x=\frac{3}{4}$

b)

$x=3$

c)

$x=6$

d)

$x=\frac{4}{3}$

79.

Solve log₂(2x - 3) = log₂(4x + 5)

a)

4

b)

-4

c)

-.33

d)

.33

80.

Condense into a single log:

\log_65+\log_62-\log_63

a)

$\log_67$

b)

$\log_64$

c)

$\log_6\frac{10}{3}$

81.

$\log_9\left(r-8\right)\ =\ 2$

a)

55

b)

$\frac{11}{6}$

c)

89

d)

6

82.

-10log₂x + 8 = -12

a)

8

b)

12

c)

1296

d)

4

83.

$6\ +\ \log_2\ x\ =\ 6$

a)

9

b)

$\frac{1}{10}$

c)

1

d)

121

84.

Solve log(x) + log(x+3) = 1

a)

5

b)

-2

c)

-5

d)

2

85.

log₂(x²- 6) = log₂(2x+2)

a)

4, -2

b)

No Solution

c)

4

d)

-2

86.

log_x1000=3

a)

1

b)

10

c)

30

d)

3

87.

log₈(4x+4)=2

a)

15

b)

12

c)

10

d)

3

88.

a)

x is undefined

b)

x = 2

c)

x= 0.4

d)

x = 33

89.

Log with a base "e" (log_e) is the same thing as...

a)

"e"

b)

Natural Logarithm (LN)

c)

Common Logarithm (Log)

d)

Natural Log, base "e"
LN_e

90.

a)

ln 8

b)

2

c)

4

d)

8

91.

Simplify the following to a single natural logarithm;
ln(2x) + ln(y) - ln(z)

a)

A

b)

B

c)

C

d)

D

92.

a)

A

b)

B

c)

C

d)

D

93.

Solve the following equation. Round your answer to two decimal places;
5 ln(x) = 7

a)

x = 4.06

b)

x = 2.04

c)

No solution

d)

None of the above

94.

Solve the following equation. Round your solution to two decimal places.
4 - 3 ln(2x + 10) = 12

a)

x = 2.20

b)

x = 1.68

c)

x = -5.00

d)

x = -4.97

95.

Solve the following for the unknown value of x. Round your solution to two decimal places.
5 ln (3x - 2) = 15

a)

x = 1.24

b)

x = 7.36

c)

x = 217,935.16

d)

x = 6.03

96.

Solve the following equation for x. Round your solution to two decimal places.
ln3x + ln2x = 3

a)

x = 12.91

b)

x = 4.02

c)

x = 3.35

d)

x = 1.83

97.

Solve the following equation for x. Round your solution to two decimal places.
ln2x - ln41 = 2

a)

x = 0.36

b)

x = 151.48

c)

x = 41

d)

None of these are solutions

98.

Which of the following is the value of x given;
e^(x + 6) = 12

a)

x = ln 6

b)

x = -6 + ln12

c)

x = ln72

d)

x = ln(-2)

99.

Which of the following is the value of x given;
7e^(3x - 5) = 49

a)

x = (ln7 + 5)/3

b)

x = ln7 + 15

c)

x = ln4

d)

x = ln7/3 + 5

100.

Solve log₉ (x - 4) + log₉ (x + 1) = log₉ (x + 5) + log₉ (x - 6)

a)

17/2

b)

-9/16

c)

13

d)

-8

101.

Change to Exponential Form:
log₆36 = 2

a)

2⁶=36

b)

6²=36

c)

36²=6

d)

36⁶=2

102.

ln (5x + 6) = 5

a)

154.413

b)

30.883

c)

147.213

d)

28.483

103.

Solve the following equation for x. Round your solution to two decimal places.

ln3+ ln2 = x

a)

x = 12.91

b)

x = 4.02

c)

x = 3.35

d)

x = 1.79

104.

ln e

a)

1

b)

0

c)

e

d)

e^e

105.

The natural logarithm has what base?

a)

10

b)

0

c)

e

d)

-e

106.

Solve the following equation for x. Round your solution to two decimal places.
ln3x + ln2x = 3

a)

x = 12.91

b)

x = 4.02

c)

x = 3.35

d)

x = 1.83

107.

ln e⁵=

a)

e⁵

b)

ln 5

c)

5

d)

5e

108.

In an exponential function, what does the 'a' represent?

a)

SLOPE

b)

RATE OF CHANGE

c)

Y-INTERCEPT

d)

COMMON RATIO

109.

Solve for x

a)

x = 1/4

b)

x = 4

c)

x = 3

d)

x = 1/3

110.

Solve:

log₂(x + 3)=5

a)

x=2

b)

x=29

c)

x=7

d)

x=32

111.

Solve the equation for x.

a)

7.389

b)

0.693

c)

0.0183

d)

6.581

112.

Solve for x:

log(x+6) = 1

a)

4

b)

4.222

c)

-5

d)

-9.550

113.

Solve for x: 3^{2x – 6} = 81

a)

x = log 4

b)

x = 5

c)

x = 4

d)

x = -1

114.

Solve for n: 7ⁿ⁺¹⁰- 8 = 6

a)

-7.374

b)

-8.644

c)

-7.360

d)

-8.853

115.

What is the inverse of f(x)=5^x+3?

a)

f^-1(x)=log₃(x-5)

b)

f^-1(x)=log₅(x)-3

c)

f^-1(x)=log₅(x-3)

d)

f^-1(x)=log₅(x+3)

116.

Determine the equation of the asymptote?

a)

x=3

b)

x=4

c)

y=-4

d)

y=4

117.

Determine the equation of the asymptote?

a)

x=5

b)

y=-5

c)

y=2

d)

x=-5

118.

Solve:

5(6)^{3x}=20

a)

2.3

b)

.26

c)

.04

d)

4

119.

Solve for x

$\log_4\left(x+5\right)+\log_4\left(7\right)=2\log_4\left(6\right)$

(a)

120.

Solve for x

$5^{\left(x-7\right)}-9=23$

Give answer as a decimal approximation (3 decimal places)

(Do not put spaces in your answer)

Remember to put the log argument in parenthesis

(a)

121.

Solve for x

$2^{\left(x+7\right)}+2=16$

Give answer as a decimal approximation (3 decimal places)

(Do not put spaces in your answer)

Remember to put the log argument in parenthesis

(a)

122.

What is the range?

a)

(-∞, ∞)

b)

(0, ∞)

c)

[1, ∞)

d)

(1, ∞)

Change of Base, Graphs of Exponentials/Logs, Properties of Logs

Total questions: 122

Worksheet time: 10hrs 31mins

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