Deck 20: Properties of Logarithms

Question 1

Rewrite the logarithm as a ratio of natural logarithms. ?
Log₅ 19
?

A) $\ln \frac { 5 } { 19 }$
B) $\ln \frac { 19 } { 5 }$
C) $\frac { \ln 5 } { \ln 19 }$
D) $\frac { \ln 19 } { \ln 5 }$
E)None of these

Answer 1

$\frac { \ln 19 } { \ln 5 }$

Question 2

Rewrite the logarithm as a ratio of common logarithms. ?
Log₅ 16
?

A) $\frac { \log 16 } { \log 5 }$
B) $\log \frac { 16 } { 5 }$
C) $\frac { \log 5 } { \log 16 }$
D) $\log \frac { 5 } { 16 }$
E)None of these

Answer 2

$\frac { \log 16 } { \log 5 }$

Question 3

Rewrite the logarithm as a ratio of natural logarithms. ?
Log_1/5 x
?

A) $\ln \frac { x } { 5 }$
B) $\frac { \ln \frac { 1 } { 5 } } { \ln x }$
C) $\frac { \ln x } { \ln \frac { 1 } { 5 } }$
D) $\ln 5 x$
E)None of these

Answer 3

$\frac { \ln x } { \ln \frac { 1 } { 5 } }$

Question 4

Use the properties of logarithms to expand the expression as a sum,difference,and/or constant multiple of logarithms.(Assume all variables are positive. ) ?
Log 5x⁴y
?

A)log 5 × 4log x × log y
B)log 5 + 4log x - log y
C)log 5 + log x + 4log y
D)log 5 + 4log x + log y
E) $\frac { \log 5 } { 4 ( \log x + \log y ) }$

Question 5

Rewrite the logarithm as a ratio of common logarithms. ?
Log_1/7 x
?

A) $\frac { \log x } { \log \frac { 1 } { 7 } }$
B) $\frac { \log \frac { 1 } { 7 } } { \log x }$
C) $\log \frac { x } { 7 }$
D) $\log 7 x$
E)None of these

Question 6

Rewrite the logarithm as a ratio of common logarithms. ?
Log_2.7 x
?

A) $\frac { \log 2.7 } { \log x }$
B)? $\frac { \log x } { \log 2.7 }$
C) $\log \frac { x } { 2.7 }$
D) $\log 2.7 x$
E)None of these

Question 7

Condense the expression to the logarithm of a single quantity. ?
Ln 8 + ln x
?

A)ln x⁸
B)ln 8x
C) $\ln \frac { 8 } { x }$
D)ln 8^x
E)ln 8 × ln x

Question 8

Use the properties of logarithms to expand the expression as a sum,difference,and/or constant multiple of logarithms.(Assume all variables are positive. ) ?
Log₃ 9x
?

A) $\frac { \log _ { 3 } 9 } { \log _ { 3 } x }$
B)log₃ 9? × log₃ x
C)log₃ 9? + log₃ x
D)log₃ 9? - log₃ x
E)None of these

Question 9

Condense the expression to the logarithm of a single quantity. ?
Log x - 3log(x + 1)
?

A)3log(x(x + 1))
B) $3 \log \frac { x } { ( x + 1 ) }$
C) $\log \frac { x } { ( x + 1 ) ^ { 3 } }$
D)log(x(x + 1)³)
E) $\frac { 1 } { 3 } \log \frac { x } { ( x + 1 ) }$

Question 10

Rewrite the logarithm as a ratio of common logarithms.? $\log _ { x } \frac { 2 } { 11 }$ ?

A) $\frac { \log x } { \log \frac { 2 } { 11 } }$
B) $\frac { \log \frac { 2 } { 11 } } { \log x }$
C) $\log \frac { 2 } { 11 } x$
D) $\log \frac { 11 } { 2 x }$
E)None of these

Question 11

Use the properties of logarithms to expand the expression as a sum,difference,and/or constant multiple of logarithms.(Assume all variables are positive. )? $\log \sqrt [ 7 ] { \frac { x ^ { 2 } } { y ^ { 4 } } }$ ?

A) $\frac { 2 } { 7 } \log x - \frac { 4 } { 7 } \log y$
B) $\frac { 1 } { 14 } \log x + \frac { 1 } { 28 } \log y$
C) $\frac { 7 } { 2 } \log x + \frac { 7 } { 4 } \log y$
D) $14 \log x + 28 \log y$
E) $\frac { 2 } { 7 } \log x + \frac { 4 } { 7 } \log y$

Question 12

Rewrite the logarithm as a ratio of natural logarithms. ?
Log_3.1 x
?

A) $\ln \frac { x } { 3.1 }$
B) $\ln 3.1 x$
C) $\frac { \ln 3.1 } { \ln x }$
D) $\frac { \ln x } { \ln 3.1 }$
E)None of these

Question 13

Use the properties of logarithms to expand the expression as a sum,difference,and/or constant multiple of logarithms.(Assume all variables are positive. ) ?
Ln 9x
?

A)ln 9 - ln x
B) $\frac { \ln 9 } { \ln x }$
C)ln 9 × ln x
D)ln 9 + ln x
E)None of these

Question 14

Condense the expression to the logarithm of a single quantity. ?
Log x - 2log y + 3log z
?

A) $\log \frac { z ^ { 3 } } { x y ^ { 2 } }$
B) $\log \frac { x } { y ^ { 2 } z ^ { 3 } }$
C) $\log \frac { x y ^ { 2 } } { z ^ { 3 } }$
D) $\log \frac { y ^ { 2 } } { x z ^ { 3 } }$
E) $$\log \frac { x z ^ { 3 } } { y ^ { 2 } }$$

Question 15

Find the exact value of the logarithmic expression without using a calculator. ​
Log₆ 216
​

A)3
B)6
C)216
D)35
E)None of these

Question 16

Rewrite the logarithm as a ratio of natural logarithms.? $\log _ { x } \frac { 5 } { 11 }$ ?

A) $\frac { \ln \frac { 5 } { 11 } } { \ln x }$
B) $\ln \frac { 11 } { 5 x }$
C) $\ln \frac { 5 } { 11 } x$
D) $\frac { \ln x } { \ln \frac { 5 } { 11 } }$
E)None of these

Question 17

Find the exact value of the logarithmic expression without using a calculator.? $\log _ { 7 } \sqrt [ 3 ] { 7 }$ ?

A)3
B) $\frac { 1 } { 3 }$
C)7
D)?21
E)None of these

Question 18

Find the exact value of the logarithmic expression without using a calculator. ​
5 ln e⁷
​

A)7
B)35
C)5
D)e
E)1

Question 19

Use the properties of logarithms to expand the expression as a sum,difference,and/or constant multiple of logarithms.(Assume all variables are positive. )? $\log _ { 8 } \frac { x ^ { 3 } } { y ^ { 3 } z ^ { 3 } }$ ?

A)3log₈ x + 3log₈ y + 3log₈ z
B)3log₈ x + 3log₈ y - 3log₈ z
C) $\frac { 3 \log _ { 8 } x } { 3 \log _ { 8 } y \times 3 \log _ { 8 } z }$
D)3log₈ x - 3log₈ y - 3log₈ z
E)3log₈ x - 3log₈ y + 3log₈ z

Question 20

Condense the expression to the logarithm of a single quantity. ?
Log₂ 10 + log₂ x
?

A)log₂ (10 - x)
B) $\ln _ { 2 } \frac { 10 } { x }$
C)log₂ (10 + x)
D)log₂ 10^x
E)log₂ 10x

Question 21

Rewrite the logarithm log₄ 17 in terms of the natural logarithm.

A) $\frac { \ln 17 } { \log _ { 4 } e }$
B) $\frac { \ln 4 } { \ln 17 }$
C) $\frac { \ln 17 } { \ln 4 }$
D)ln 17
E)ln 4 ln 17

Question 22

Find the exact value of $\log _ { 7 } \sqrt [ 3 ] { 49 }$ without using a calculator.

A) $\frac { 3 } { 49 }$
B) $\frac { 14 } { 3 }$
C) $\frac { 2 } { 3 }$
D)-1
E) $\frac { 49 } { 3 }$

Question 23

Condense the expression to the logarithm of a single quantity.? $\frac { 1 } { 2 } \left[ \log _ { 9 } ( x + 6 ) + 2 \log _ { 9 } ( x - 6 ) \right] - 12 \log _ { 9 } x$ ?

A) $\log _ { 9 } \frac { ( x + 6 ) \sqrt { x - 6 } } { x ^ { 6 } }$
B) $\log _ { 9 } \frac { x ^ { 12 } \sqrt { x - 6 } } { ( x + 6 ) }$
C) $\log _ { 9 } \frac { x ^ { 6 } \sqrt { x - 6 } } { ( x + 6 ) }$
D) $\log _ { 9 } \frac { x ^ { 12 } \sqrt { x + 6 } } { ( x - 6 ) }$
E) $\log _ { 9 } \frac { ( x - 6 ) \sqrt { x + 6 } } { x ^ { 12 } }$

Question 24

Evaluate the logarithm using the change-of-base formula.Round your result to three decimal places.
​
Log₁₅ 1,500
​

A)5.701
B)3.701
C)2.701
D)6.701
E)4.701

Question 25

Evaluate the logarithm using the change-of-base formula.Round your result to three decimal places.
​
Log₂ 7
​

A)2.807
B)6.807
C)4.807
D)5.807
E)3.807

Question 26

Rewrite the logarithm log₆ 412 in terms of the common logarithm (base 10).

A)log 6 log 412
B) $\frac { \log 412 } { \log 6 }$
C) $\frac { \log 6 } { \log 412 }$
D)log 412
E) $\frac { \log 412 } { \log _ { 6 } 10 }$

Question 27

Use a graphing utility to graph the functions given by ?
Y₁ = ln x - ln (x - 4)
?
And? $$y _ { 2 } = \ln \frac { x } { x - 4 }$$ ?
In the same viewing window.
?

A)
B)
C)
D)
E)

Question 28

Use the change-of-base formula to rewrite the logarithm as a ratio of logarithms.Then use a graphing utility to graph the ratio.
?
F(x)= log₄ x
?

A)f(x)= log x + log 4 = ln x + ln 4 ?
B) $$f ( x ) = \frac { \log 4 } { \log x } = \frac { \ln 4 } { \ln x }$$
C) $$f ( x ) = \log \frac { x } { 4 } = \ln \frac { x } { 4 }$$
D) $$f ( x ) = \log \frac { 4 } { x } = \ln \frac { 4 } { x }$$
E) $$f ( x ) = \frac { \log x } { \log 4 } = \frac { \ln x } { \ln 4 }$$ ?

Question 29

Evaluate the logarithm using the change-of-base formula.Round your result to three decimal places.? $\log _ { \frac { 1 } { 3 } } 6$ ?

A)-0.631
B)2.369
C)0.369
D)-1.631
E)1.369

Question 30

Use the change-of-base formula to rewrite the logarithm as a ratio of logarithms.Then use a graphing utility to graph the ratio.
?
F(x)= log_11.7 x
?

A) $$f ( x ) = \frac { \log x } { \log 11.7 } = \frac { \ln x } { \ln 11.7 }$$
B) $$f ( x ) = \frac { \log 11.7 } { \log x } = \frac { \ln 11.7 } { \ln x }$$
C)f(x)= log x? + log 11.7 = ln x? + ln 11.7
D) $$f ( x ) = \log \frac { 11.7 } { x } = \ln \frac { 11.7 } { x }$$
E) $$f ( x ) = \log \frac { x } { 11.7 } = \ln \frac { x } { 11.7 }$$ ?

Question 31

Use the change-of-base formula to rewrite the logarithm as a ratio of logarithms.Then use a graphing utility to graph the ratio.? $$f ( x ) = \log _ { \frac { 1 } { 5 } } x$$ ?

A) $$f ( x ) = \frac { \log \frac { 1 } { 5 } } { \log x } = \frac { \ln \frac { 1 } { 5 } } { \ln x }$$
B) $$f ( x ) = \frac { \log x } { \log \frac { 1 } { 5 } } = \frac { \ln x } { \ln \frac { 1 } { 5 } }$$
C) $$f ( x ) = \log x + \log \frac { 1 } { 5 } = \ln x + \ln \frac { 1 } { 5 }$$
D) $$f ( x ) = \log \frac { \frac { 1 } { 5 } } { x } = \ln \frac { \frac { 1 } { 5 } } { x }$$
E) $$f ( x ) = \log \frac { x } { \frac { 1 } { 5 } } = \ln \frac { x } { \frac { 1 } { 5 } }$$

Question 32

Use the properties of logarithms to rewrite and simplify the logarithmic expression. ​
Ln (3e⁴)
​

A)ln 4
B)ln 3 + 4
C)ln 4 + 3
D)ln 7
E)ln 3

Question 33

Condense the expression to the logarithm of a single quantity.? $\ln x - [ \ln ( x + 6 ) + \ln ( x - 6 ) ]$ ?

A) $\ln \frac { ( x - 6 ) } { x ( x + 6 ) }$
B) $$\ln \frac { ( x + 6 ) } { x ( x - 6 ) }$$
C) $\ln \frac { x ( x - 6 ) } { ( x + 6 ) }$
D) $\ln \frac { x } { ( x - 6 ) ( x + 6 ) }$
E) $\ln \frac { x ( x + 6 ) } { ( x - 6 ) }$

Question 34

Evaluate the logarithm log₇ 126 using the change of base formula.Round to 3 decimal places.

A)4.836
B)2.485
C)0.402
D)9.411
E)2.1

Question 35

Simplify the expression log₅ 175.

A)35log₅ 2
B)2 log₅ 7
C)7
D)​2 + log₅ 7
E)The expression cannot be simplified.

Question 36

Use the properties of logarithms to rewrite and simplify the logarithmic expression. ​
Log₂ (4² · 7⁴⁾
​

A)4 + 9 log₂ 4
B)9 + 4 log₂ 4
C)2 + 2 log₂ 7
D)4 + 4 log₂ 7
E)2 + 4 log₂ 7

Question 37

Evaluate the logarithm log_1/3 0.603 using the change of base formula.Round to 3 decimal places.

A)2.172
B)0.556
C)-0.506
D)-0.22
E)0.46

Question 38

Evaluate the logarithm using the change-of-base formula.Round your result to three decimal places. ​
Log₉ 0.1
​

A)2.952
B)-0.048
C)-1.048
D)1.952
E)0.952

Question 39

Condense the expression to the logarithm of a single quantity.? $2 [ 3 \ln x - \ln ( x + 8 ) - \ln ( x - 8 ) ]$ ?

A) $\ln \left( \frac { x ^ { 3 } } { ( x - 8 ) ( x + 8 ) } \right) ^ { 2 }$
B) $\ln \left( \frac { x ^ { 3 } ( x + 8 ) } { ( x - 8 ) } \right) ^ { 2 }$
C) $\ln \left( \frac { x ^ { 3 } ( x - 8 ) } { ( x + 8 ) } \right) ^ { 2 }$
D) $\ln \left( \frac { ( x - 8 ) } { x ^ { 3 } ( x + 8 ) } \right) ^ { 2 }$
E) $\ln \left( \frac { ( x + 8 ) } { x ^ { 3 } ( x - 8 ) } \right) ^ { 2 }$

Question 40

Use the properties of logarithms to rewrite and simplify the logarithmic expression.? $\ln \left( \frac { 2 } { e ^ { 3 } } \right)$ ?

A)ln 2
B)ln 5
C)ln 2 - 3
D)ln 3 - 2
E)ln 3

Question 41

Assume that x,y,and z are positive numbers.Use the properties of logarithms to write the expression $- 2 \log _ { b } x - 6 \log _ { b } y + \frac { 1 } { 7 } \log _ { b } z$ as the logarithm of one quantity. ?

A) $\log _ { b } \frac { z ^ { 1 / 7 } } { x ^ { 6 } y ^ { 2 } }$
B) $\log _ { b } \frac { z ^ { 1 / 6 } } { x ^ { 2 } y ^ { 7 } }$
C) $\log _ { b } \frac { z ^ { 1 / 2 } } { x ^ { 7 } y ^ { 6 } }$
D) $\log _ { b } \frac { x ^ { 1 / 7 } } { y ^ { 2 } z ^ { 6 } }$
E) $\log _ { b } \frac { z ^ { 1 / 7 } } { x ^ { 2 } y ^ { 6 } }$

Question 42

Assume that x is a positive number.Use the properties of logarithms to write the expression ?
Log_b (x + 8)- log_b x as the logarithm of one quantity.
?

A) $\log _ { b } \frac { x ^ { 2 } + 8 } { x }$
B) $\log _ { b } \frac { x + 8 } { x }$
C) $\log _ { b } \frac { x - 8 } { x }$
D) $\log _ { b } \frac { x ^ { 2 } + 8 } { 8 }$
E)log_b (x² - 8x)

Question 43

Condense the expression 7(log x - log y)to the logarithm of a single term.

A) $\log \left( \frac { x } { y } \right) ^ { 7 }$
B) $\log \frac { 7 x } { 7 y }$
C) $\log \frac { x ^ { 7 } } { y }$
D) $\log \frac { x ^ { 7 } } { \sqrt [ 7 ] { y } }$
E)7(log x - log y)

Question 44

Assume that x,y,and c are positive numbers.Use the properties of logarithms to write the expression $\log _ { c } 5 x y$ in terms of the logarithms of x and y. ?

A) $\log _ { c } 5 + \log _ { c } x y$
B) $\log _ { c } 5 + \log _ { c } x + \log _ { c } y$
C) $\log _ { c } 5 + \log _ { c } x$
D) $\log _ { c } 5 + \log _ { 5 } x + \log _ { 5 } y$
E) $\log _ { c } 5 + \log _ { c } y$

Question 45

Simplify the expression $\log _ { 3 } \left( \frac { 1 } { 27 } \right) ^ { 3 }$ .

A)3
B)0
C)-9
D)-81
E)The expression cannot be simplified.

Question 46

Condense the expression $$\frac { 1 } { 3 } \left[ \log _ { 6 } x + \log _ { 6 } 7 \right] - \left[ \log _ { 6 } y \right]$$ to the logarithm of a single term.

A) $\log _ { 6 } \frac { ( 7 x ) ^ { 3 } } { y }$
B) $\log _ { 6 } \frac { \sqrt [ 3 ] { 7 x } } { y }$
C) $\log _ { 6 } \sqrt [ 3 ] { \frac { 7 x } { y } }$
D) $\log _ { 6 } \sqrt [ 3 ] { 7 x } - \log _ { 6 } y$
E) $\log _ { 6 } \frac { 7 x } { 3 y }$

Question 47

Condense the expression 3(log x - log y)to the logarithm of a single term.

A) $\log \frac { x ^ { 3 } } { y }$
B) $\log \frac { 3 x } { 3 y }$
C) $\log \frac { x } { y ^ { 3 } }$
D)3(log x - log y)
E) $\log \left( \frac { x } { y } \right) ^ { 3 }$

Question 48

Find the exact value of $\ln e ^ { 250 } - \ln \sqrt { e }$ without using a calculator.

A)5
B)2.5
C)1.25
D)2
E)3

Question 49

Assume that x,y,and c are positive numbers.Use the properties of logarithms to write the expression $\log _ { c } \sqrt [ 6 ] { x y }$ in terms of the logarithms of x and y. ?

A) $\frac { 1 } { 6 } \log _ { c } x + \log _ { c } y$
B) $6 \log _ { c } x + 6 \log _ { c } y$
C) $\frac { 1 } { 6 } \log _ { c } ( x + y )$
D) $\log _ { c } x + \log _ { c } y$
E) $\frac { 1 } { 6 } \log _ { c } x + \frac { 1 } { 6 } \log _ { c } y$

Question 50

Assume that x,y and c are positive numbers.Use the properties of logarithms to write the expression $\log _ { c } x ^ { 6 } y ^ { 4 }$ in terms of the logarithms of x and y. ?

A) $6 \log _ { c } x + 4 \log _ { c } y$
B) $24 \log _ { c } x + y$
C) $6 \log _ { c } x + 6 \log _ { c } y$
D) $24 \log _ { c } x + 4 \log _ { c } y$
E) $6 \log _ { c } x + 24 \log _ { c } y$

Question 51

Assume that x,y,z and b are positive numbers.Use the properties of logarithms to write the expression $\log _ { b } \sqrt [ 4 ] { \frac { x ^ { 5 } y ^ { 2 } } { z ^ { 4 } } }$ in terms of the logarithms of x,y,and z. ?

A) $\frac { 5 } { 4 } \log _ { b } x + \log _ { b } y - \log _ { b } z$
B) $\frac { 5 } { 4 } \log _ { b } x + \frac { 1 } { 2 } \log _ { b } y - \log _ { b } z$
C) $\log _ { b } x + \frac { 1 } { 2 } \log _ { b } y - \log _ { b } z$
D) $\frac { 5 } { 4 } \log _ { b } ( x + y + z )$
E) $20 \log _ { b } x + 8 \log _ { b } y - 16 \log _ { b } z$

Question 52

Put the expressions in the appropriate order: $\frac { \ln 8 } { \ln e } , \frac { \ln 8 } { e } , \ln 8$ .

A) $\ln 8 < \frac { \ln 8 } { e } < \frac { \ln 8 } { \ln e }$
B) $\frac { \ln 8 } { e } < \frac { \ln 8 } { \ln e } = \ln 8$
C)? $\ln 8 < \frac { \ln 8 } { e } = \frac { \ln 8 } { \ln e }$
D) $\frac { \ln 8 } { \ln e } < \ln 8 < \frac { \ln 8 } { \ln e }$
E)The expressions are equivalent.

Question 53

Condense the expression $\frac { 1 } { 5 } \left[ \log _ { 4 } x + \log _ { 4 } 3 \right] - \left[ \log _ { 4 } y \right]$ to the logarithm of a single term.

A) $\log _ { 4 } \frac { ( 3 x ) ^ { 5 } } { y }$
B) $\log _ { 4 } \frac { 3 x } { 5 y }$
C) $\log _ { 4 } \sqrt [ 5 ] { \frac { 3 x } { y } }$
D) $\log _ { 4 } \frac { \sqrt [ 5 ] { 3 x } } { y }$
E) $\log _ { 4 } \sqrt [ 5 ] { 3 x } - \log _ { 4 } y$

Question 54

Find the exact value of $\log _ { 7 } \sqrt [ 3 ] { 49 }$ without using a calculator.

A) $\frac { 49 } { 3 }$
B) $\frac { 3 } { 49 }$
C)? $\frac { 14 } { 3 }$
D) $\frac { 2 } { 3 }$
E)-1

Question 55

The pH of an acidic solution is a measure of the concentration of the acid particles in the solution,with smaller values of the pH indicating higher acid concentration.In a laboratory experiment,the pH of a certain acid solution is changed by dissolving over-the-counter antacid tablets into the solution.In this experiment,the pH changes according to the equation $\mathrm { pH } = 5.14 + \log \left( \frac { x } { 0.40 - x } \right)$ , where x is the number of grams of antacid added to the solution.What is the pH of the solution after the addition of 0.05 grams of antacid tablet?

A)5.99
B)4.02
C)-0.85
D)4.29
E)5.14

Question 56

Condense the expression log₄ x + log₄ 3 to the logarithm of a single term.

A)log₄ -3x
B)log₄ 3x
C)log₄ x³
D)log(3x)⁴
E)log₄ (x + 3)

Question 57

Find the exact value of $\log _ { 4 } 44 - \log _ { 4 } 11$ without using a calculator.

A)1
B)4
C)11
D) $\frac { 11 } { 2 }$
E) $\frac { 1 } { 2 }$