Question 1

Evaluate the following. $$\log _ { b } 1$$

Accepted Answer

A) 0 
B)  $b$ 
C) 1 
D)  $\frac { 1 } { b }$ 
E) none of these 
A) 0 
B)  $b$ 
C) 1 
D)  $\frac { 1 } { b }$ 
E) none of these

Question 2

Graph $y = \log _ { 6 } x$ by reflecting the graph of $g ( x ) = 6 ^ { x }$ across the line y = x .

Accepted Answer

A)    
B)    
C)    
A)    
B)    
C)

Question 3

The half-life of radium is approximately 1,600 years. If the present amount of radium in a certain location is 900 grams, how much will remain after 400 years? Express your answer to the nearest gram.

Accepted Answer

A) Q = 734 gram 
B) Q = 768 gram 
C) Q = 757 gram 
D) Q = 752 gram 
E) Q = 744 gram 
A) Q = 734 gram 
B) Q = 768 gram 
C) Q = 757 gram 
D) Q = 752 gram 
E) Q = 744 gram

Question 4

Use a calculator and the change-of-base formula to find the logarithm to four decimal places. $$\log _ { \frac { 1 } { 3 } } 3$$

Accepted Answer

A) 0.5769 
B) - 1.0000 
C) 1.1005 
D) - 1.0078 
E) 1.7654 
A) 0.5769 
B) - 1.0000 
C) 1.1005 
D) - 1.0078 
E) 1.7654

Question 5

Solve the exponential equation and express approximate solutions to the nearest hundredth. e x - 4 = 38

Accepted Answer

A) x = 7.11 
B) x = 4.00 
C) x = 8.17 
D) x = 7.64 
E) x = 7.52 
A) x = 7.11 
B) x = 4.00 
C) x = 8.17 
D) x = 7.64 
E) x = 7.52

Question 6

Graph the exponential function. $$f ( x ) = e ^ { - x }$$

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 7

Graph $y = \log _ { \frac { 1 } { 4 } } x$ by graphing $\left( \frac { 1 } { 4 } \right) ^ { y } = x$ .

Accepted Answer

A)    
B)    
C)    
A)    
B)    
C)

Question 8

Solve the logarithmic equation. Check the solution. $$\log ( 8 x + 5 ) - \log ( 7 x + 8 ) = 0$$

Accepted Answer

A) x = 1 
B) x = 0 
C) x = 3 
D) x = 4 
A) x = 1 
B) x = 0 
C) x = 3 
D) x = 4

Question 9

Solve the logarithmic equation. Check the solution. $$\log \left( x ^ { 2 } + 7 x \right) = \log \left( x ^ { 2 } + 56 \right)$$

Accepted Answer

A) x = 9 
B) x = 8 
C) x = 11 
D) x = 10 
A) x = 9 
B) x = 8 
C) x = 11 
D) x = 10

Question 10

Solve the equation log 6  x = 2.

Accepted Answer

A) x = 64 
B) x = 42 
C) x = 2 
D) x = 6 
E) x = 36 
A) x = 64 
B) x = 42 
C) x = 2 
D) x = 6 
E) x = 36

Question 11

Assume that $x$ , $y$ , $Z$ , and $b ( b \neq 1 )$ are positive numbers Use the properties of logarithms to write the expression in terms of the logarithms of $x$ , $y$ , and $Z$ . $\log _ { b } x y ^ { 5 } z ^ { 6 }$

Accepted Answer

A)  $\log _ { b } x + \frac { \log _ { b } y } { 5 } + \frac { \log _ { b } z } { 6 }$ 
B)  $\log _ { b } x + \left( \log _ { b } y \right) ^ { 5 } + \left( \log _ { b } z \right) ^ { 6 }$ 
C)  $\log _ { b } x + 5 \log _ { b } y + 6 \log _ { b } z$ 
D)  $- \log _ { b } x - 5 \log _ { b } y - 6 \log _ { b } z$ 
E)  $\log _ { b } x - \frac { \log _ { b } y } { 5 } - \frac { \log _ { b } z } { 6 }$ 
A)  $\log _ { b } x + \frac { \log _ { b } y } { 5 } + \frac { \log _ { b } z } { 6 }$ 
B)  $\log _ { b } x + \left( \log _ { b } y \right) ^ { 5 } + \left( \log _ { b } z \right) ^ { 6 }$ 
C)  $\log _ { b } x + 5 \log _ { b } y + 6 \log _ { b } z$ 
D)  $- \log _ { b } x - 5 \log _ { b } y - 6 \log _ { b } z$ 
E)  $\log _ { b } x - \frac { \log _ { b } y } { 5 } - \frac { \log _ { b } z } { 6 }$

Question 12

Use a calculator and the change-of-base formula to find the logarithm to four decimal places. $$\log _ { 3 } \pi$$

Accepted Answer

A) 0.6189 
B) 1.0420 
C) 1.8074 
D) 1.0342 
E) 0.9415 
A) 0.6189 
B) 1.0420 
C) 1.8074 
D) 1.0342 
E) 0.9415

Question 13

Use your calculator to find x when given ln x . Express answer to four decimal places. $$\ln x = 2.5138$$

Accepted Answer

A) 12.4018 
B) 12.3418 
C) 12.3818 
D) 12.3518 
E) 12.3718 
A) 12.4018 
B) 12.3418 
C) 12.3818 
D) 12.3518 
E) 12.3718

Question 14

Find the value of $x$ . $\log _ { 3 } 81 = x$

Accepted Answer

A) 3 
B) 5 
C) 6 
D) 2 
E) 4 
A) 3 
B) 5 
C) 6 
D) 2 
E) 4

Question 15

Assume that $x$ , $y$ , $z$ , and $\text { a }$ $( b \neq 1 )$ are positive numbers.Use the properties of logarithms to write the expression in terms of the logarithms of x , y , and z . $\log _ { b } \frac { x } { y z }$

Accepted Answer

A) log b  x log b  y log b  z 
B) log b  x + log b  y + log b  z 
C) - log b  x - log b  y - log b  z 
D) log b  x - log b  y - log b  z 
A) log b  x log b  y log b  z 
B) log b  x + log b  y + log b  z 
C) - log b  x - log b  y - log b  z 
D) log b  x - log b  y - log b  z

Question 16

Solve the equation log 3 (2 x - 1)- log 3 ( x - 2)= 1.

Accepted Answer

A) x = 4 
B) x = 5 
C) x = 2 
D) x = 7 
E) x = 6 
A) x = 4 
B) x = 5 
C) x = 2 
D) x = 7 
E) x = 6

Question 17

Graph the exponential function. $$f ( x ) = \left( \frac { 1 } { 3 } \right) ^ { x }$$

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Evaluate the following. $\log _ { b } 1$

Graph $y = \log _ { 6 } x$ by reflecting the graph of $g ( x ) = 6 ^ { x }$ across the line y = x .

The half-life of radium is approximately 1,600 years. If the present amount of radium in a certain location is 900 grams, how much will remain after 400 years? Express your answer to the nearest gram.

Use a calculator and the change-of-base formula to find the logarithm to four decimal places. $\log _ { \frac { 1 } { 3 } } 3$

Solve the exponential equation and express approximate solutions to the nearest hundredth. e^{x - 4} = 38

Graph the exponential function. $f ( x ) = e ^ { - x }$

Graph $y = \log _ { \frac { 1 } { 4 } } x$ by graphing $\left( \frac { 1 } { 4 } \right) ^ { y } = x$ .

Solve the logarithmic equation. Check the solution. $\log ( 8 x + 5 ) - \log ( 7 x + 8 ) = 0$

Solve the logarithmic equation. Check the solution. $\log \left( x ^ { 2 } + 7 x \right) = \log \left( x ^ { 2 } + 56 \right)$

Solve the equation log₆ x = 2.

Assume that $x$ , $y$ , $Z$ , and $b ( b \neq 1 )$ are positive numbers Use the properties of logarithms to write the expression in terms of the logarithms of $x$ , $y$ , and $Z$ . $\log _ { b } x y ^ { 5 } z ^ { 6 }$

Use a calculator and the change-of-base formula to find the logarithm to four decimal places. $\log _ { 3 } \pi$

Use your calculator to find x when given ln x . Express answer to four decimal places. $\ln x = 2.5138$

Find the value of $x$ . $\log _ { 3 } 81 = x$

Assume that $x$ , $y$ , $z$ , and $\text { a }$ $( b \neq 1 )$ are positive numbers.Use the properties of logarithms to write the expression in terms of the logarithms of x , y , and z . $\log _ { b } \frac { x } { y z }$

Solve the equation log₃ (2 x - 1)- log₃ ( x - 2)= 1.

Graph the exponential function. $f ( x ) = \left( \frac { 1 } { 3 } \right) ^ { x }$

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Exam 11: Exponential and Logarithmic Functions

Evaluate the following. log⁡b1\log _ { b } 1logb​1

Graph y=log⁡6xy = \log _ { 6 } xy=log6​x by reflecting the graph of g(x)=6xg ( x ) = 6 ^ { x }g(x)=6x across the line y = x .

The half-life of radium is approximately 1,600 years. If the present amount of radium in a certain location is 900 grams, how much will remain after 400 years? Express your answer to the nearest gram.

Use a calculator and the change-of-base formula to find the logarithm to four decimal places. log⁡133\log _ { \frac { 1 } { 3 } } 3log31​​3

Solve the exponential equation and express approximate solutions to the nearest hundredth. e x - 4 = 38

Graph the exponential function. f(x)=e−xf ( x ) = e ^ { - x }f(x)=e−x

Graph y=log⁡14xy = \log _ { \frac { 1 } { 4 } } xy=log41​​x by graphing (14)y=x\left( \frac { 1 } { 4 } \right) ^ { y } = x(41​)y=x .

Solve the logarithmic equation. Check the solution. log⁡(8x+5)−log⁡(7x+8)=0\log ( 8 x + 5 ) - \log ( 7 x + 8 ) = 0log(8x+5)−log(7x+8)=0

Solve the logarithmic equation. Check the solution. log⁡(x2+7x)=log⁡(x2+56)\log \left( x ^ { 2 } + 7 x \right) = \log \left( x ^ { 2 } + 56 \right)log(x2+7x)=log(x2+56)

Solve the equation log 6 x = 2.

Assume that xxx , yyy , ZZZ , and b(b≠1)b ( b \neq 1 )b(b=1) are positive numbers Use the properties of logarithms to write the expression in terms of the logarithms of xxx , yyy , and ZZZ . log⁡bxy5z6\log _ { b } x y ^ { 5 } z ^ { 6 }logb​xy5z6

Use a calculator and the change-of-base formula to find the logarithm to four decimal places. log⁡3π\log _ { 3 } \pilog3​π

Use your calculator to find x when given ln x . Express answer to four decimal places. ln⁡x=2.5138\ln x = 2.5138lnx=2.5138

Find the value of xxx . log⁡381=x\log _ { 3 } 81 = xlog3​81=x

Assume that xxx , yyy , zzz , and a \text { a } a (b≠1)( b \neq 1 )(b=1) are positive numbers.Use the properties of logarithms to write the expression in terms of the logarithms of x , y , and z . log⁡bxyz\log _ { b } \frac { x } { y z }logb​yzx​

Solve the equation log 3 (2 x - 1)- log 3 ( x - 2)= 1.

Graph the exponential function. f(x)=(13)xf ( x ) = \left( \frac { 1 } { 3 } \right) ^ { x }f(x)=(31​)x

Basic Concepts and Properties

Equations, Inequalities, and Problem Solving

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Evaluate the following. $\log _ { b } 1$

Graph $y = \log _ { 6 } x$ by reflecting the graph of $g ( x ) = 6 ^ { x }$ across the line y = x .

Use a calculator and the change-of-base formula to find the logarithm to four decimal places. $\log _ { \frac { 1 } { 3 } } 3$

Solve the exponential equation and express approximate solutions to the nearest hundredth. e^{x - 4} = 38

Graph the exponential function. $f ( x ) = e ^ { - x }$

Graph $y = \log _ { \frac { 1 } { 4 } } x$ by graphing $\left( \frac { 1 } { 4 } \right) ^ { y } = x$ .

Solve the logarithmic equation. Check the solution. $\log ( 8 x + 5 ) - \log ( 7 x + 8 ) = 0$

Solve the logarithmic equation. Check the solution. $\log \left( x ^ { 2 } + 7 x \right) = \log \left( x ^ { 2 } + 56 \right)$

Solve the equation log₆ x = 2.

Assume that $x$ , $y$ , $Z$ , and $b ( b \neq 1 )$ are positive numbers Use the properties of logarithms to write the expression in terms of the logarithms of $x$ , $y$ , and $Z$ . $\log _ { b } x y ^ { 5 } z ^ { 6 }$

Use a calculator and the change-of-base formula to find the logarithm to four decimal places. $\log _ { 3 } \pi$

Use your calculator to find x when given ln x . Express answer to four decimal places. $\ln x = 2.5138$

Find the value of $x$ . $\log _ { 3 } 81 = x$

Assume that $x$ , $y$ , $z$ , and $\text { a }$ $( b \neq 1 )$ are positive numbers.Use the properties of logarithms to write the expression in terms of the logarithms of x , y , and z . $\log _ { b } \frac { x } { y z }$

Solve the equation log₃ (2 x - 1)- log₃ ( x - 2)= 1.

Graph the exponential function. $f ( x ) = \left( \frac { 1 } { 3 } \right) ^ { x }$