Question 1

Identify the vertical asymptote of the function $f ( x ) = 2 + \log ( x + 3 )$.

Accepted Answer

A)  $x = 0$ 
B)  $x = - 2$ 
C)  $x = - 3$ 
D)  $x = 3$ 
E)  The function has no vertical asymptote. 
A)  $x = 0$ 
B)  $x = - 2$ 
C)  $x = - 3$ 
D)  $x = 3$ 
E)  The function has no vertical asymptote. C

Question 2

Simplify the expression below.

Accepted Answer

A)  $6 + e ^ { 4 }$ 
B)  $6 + x ^ { 4 }$ 
C)  $6 + \ln 4$ 
D)  $\ln 10$ 
E)  $6 + 4 \ln x$ 
A)  $6 + e ^ { 4 }$ 
B)  $6 + x ^ { 4 }$ 
C)  $6 + \ln 4$ 
D)  $\ln 10$ 
E)  $6 + 4 \ln x$ B

Question 3

Determine which $x$-value below is a solution of the equation $\log _ { 7 } \left( \frac { 5 } { 3 } x \right) = 3$.
$$x = \frac { 21 } { 125 } , x = \frac { 1029 } { 5 } , x = \frac { 189 } { 5 } , x = 21 , x = 15$$

Accepted Answer

A)  $x = \frac { 21 } { 125 }$ 
B)  $x = \frac { 1029 } { 5 }$ 
C)  $x = \frac { 189 } { 5 }$ 
D)  $x = 21$ 
E)  $x = 15$ 
A)  $x = \frac { 21 } { 125 }$ 
B)  $x = \frac { 1029 } { 5 }$ 
C)  $x = \frac { 189 } { 5 }$ 
D)  $x = 21$ 
E)  $x = 15$ B

Question 4

Identify the $x$-intercept of the function $y = 3 + \log _ { 4 } x$.

Accepted Answer

A)  64 
B)  $\frac { 1 } { 64 }$ 
C)  $- 3$ 
D)  12 
E)  The function has no $x$-intercept. 
A)  64 
B)  $\frac { 1 } { 64 }$ 
C)  $- 3$ 
D)  12 
E)  The function has no $x$-intercept.

Question 5

Evaluate the logarithm $\log _ { 1 / 3 } 0.113$ using the change of base formula. Round to 3 decimal places.

Accepted Answer

A)  $- 2.180$ 
B)  $2.395$ 
C)  $0.504$ 
D)  $1.985$ 
E)  $- 0.947$ 
A)  $- 2.180$ 
B)  $2.395$ 
C)  $0.504$ 
D)  $1.985$ 
E)  $- 0.947$

Question 6

Solve the equation below algebraically. Round your result to three decimal places.
$$\frac { 1 + 2 \ln x } { x ^ { 5 } } = 0$$

Accepted Answer

A)  $0.862$ 
B)  $- 0.821$ 
C)  $1.855$ 
D)  $- 1.217$ 
E)  $\quad 0.607$ 
A)  $0.862$ 
B)  $- 0.821$ 
C)  $1.855$ 
D)  $- 1.217$ 
E)  $\quad 0.607$

Question 7

Solve the equation below algebraically.
$$- 2 x ^ { 2 } e ^ { 4 x } - 16 x e ^ { 4 x } = 0$$

Accepted Answer

A)  $- 8$ and 4 
B)  4 and 5 
C)  $- 8$ and 5 
D)  0 and 5 
E)  $- 8$ and 0 
A)  $- 8$ and 4 
B)  4 and 5 
C)  $- 8$ and 5 
D)  0 and 5 
E)  $- 8$ and 0

Question 8

Solve the exponential equation below algebraically. Round your result to three decimal places.

Accepted Answer

A)  $1.099$
$$e ^ { 2 x } + 3 e ^ { x } - 18 = 0$$ 
B)  $0.214$ 
C)  $2.077$ 
D)  $- 0.188$ 
E)  $3.271$ 
A)  $1.099$
$$e ^ { 2 x } + 3 e ^ { x } - 18 = 0$$ 
B)  $0.214$ 
C)  $2.077$ 
D)  $- 0.188$ 
E)  $3.271$

Question 9

Identify the graph of the function.
$$f ( x ) = \left( \frac { 1 } { 2 } \right) ^ { x }$$

Accepted Answer

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

Question 10

What is the value of the function $f ( x ) = 3.8 e ^ { - 1.5 x }$ at $x = 2.5 ?$ Round to 3 decimal places.

Accepted Answer

A)  $0.848$ 
B)  $0.089$ 
C)  $10.329$ 
D)  $- 38.736$ 
E)  $0.001$ 
A)  $0.848$ 
B)  $0.089$ 
C)  $10.329$ 
D)  $- 38.736$ 
E)  $0.001$

Question 11

Condense the expression below to the logarithm of a single quantity.
$$\frac { 3 } { 2 } \log _ { 7 } ( z + 5 )$$

Accepted Answer

A)  $\log _ { 7 } ( \sqrt [ 3 ] { z } + 5 )$ 
B)  $\log _ { 7 } \left( \sqrt { z ^ { 3 } } + 5 \right)$ 
C)  $\log _ { 7 } \sqrt [ 3 ] { z + 5 }$ 
D)  $\log _ { 7 } \frac { 3 ( z + 5 ) } { 2 }$ 
E)  $\quad \log _ { 7 } \sqrt { ( z + 5 ) ^ { 3 } }$ 
A)  $\log _ { 7 } ( \sqrt [ 3 ] { z } + 5 )$ 
B)  $\log _ { 7 } \left( \sqrt { z ^ { 3 } } + 5 \right)$ 
C)  $\log _ { 7 } \sqrt [ 3 ] { z + 5 }$ 
D)  $\log _ { 7 } \frac { 3 ( z + 5 ) } { 2 }$ 
E)  $\quad \log _ { 7 } \sqrt { ( z + 5 ) ^ { 3 } }$

Question 12

Solve $\ln x ^ { 2 } = 3$ for $x$.

Accepted Answer

A)  $e ^ { 9 }$ 
B)  $10 ^ { \sqrt { 3 } }$ 
C)  $e ^ { \sqrt { 3 } }$ 
D)  $- e ^ { 3 / 2 } , e ^ { 3 / 2 }$ 
E)  no solution 
A)  $e ^ { 9 }$ 
B)  $10 ^ { \sqrt { 3 } }$ 
C)  $e ^ { \sqrt { 3 } }$ 
D)  $- e ^ { 3 / 2 } , e ^ { 3 / 2 }$ 
E)  no solution

Question 13

What is the value of the function $f ( x ) = 3.3 e ^ { - 1.8 x }$ at $x = 2.5 ?$ Round to 3 decimal places.

Accepted Answer

A)  $0.545$ 
B)  $0.037$ 
C)  $6.645$ 
D)  $- 40.366$ 
E)  $0.000$ 
A)  $0.545$ 
B)  $0.037$ 
C)  $6.645$ 
D)  $- 40.366$ 
E)  $0.000$

Question 14

Find the exponential model $y = a e ^ { b x }$ that fits the points shown in the table below. Round parameters to the nearest thousandth.
$\begin{array}{|c|c|c|}
\hline x & 0 & 1 \
\hline y & 5 & 10 \
\hline
\end{array}$

Accepted Answer

A)  $y = 5 e ^ { 0.693 x }$ 
B)  $y = 0.693 e ^ { - 5 x }$ 
C)  $y = - 5 e ^ { - 0.693 x }$ 
D)  $y = 5 e ^ { - 0.693 x }$ 
E)  $y = 0.693 e ^ { 5 x }$ 
A)  $y = 5 e ^ { 0.693 x }$ 
B)  $y = 0.693 e ^ { - 5 x }$ 
C)  $y = - 5 e ^ { - 0.693 x }$ 
D)  $y = 5 e ^ { - 0.693 x }$ 
E)  $y = 0.693 e ^ { 5 x }$

Question 15

Find the exact value of the logarithm without using a calculator, if possible.
$$\ln \frac { 1 } { \sqrt [ 2 ] { e } }$$

Accepted Answer

A)  $- 2$ 
B)  2 
C)  $- \frac { 1 } { 2 }$ 
D)  $\frac { 1 } { 2 }$ 
E)  not possible without using a calculator 
A)  $- 2$ 
B)  2 
C)  $- \frac { 1 } { 2 }$ 
D)  $\frac { 1 } { 2 }$ 
E)  not possible without using a calculator

Question 16

Solve the logarithmic equation below algebraically.
$$\log _ { 6 } ( 2 x + 4 ) = \log _ { 6 } ( 5 x + 1 )$$

Accepted Answer

A)  3 
B)  $- 2$ 
C)  $- 4$ 
D)  2 
E)  1 
A)  3 
B)  $- 2$ 
C)  $- 4$ 
D)  2 
E)  1

Question 17

Rewrite the logarithmic equation $\log _ { 4 } \frac { 1 } { 16 } = - 2$ in exponential form.

Accepted Answer

A)  $4 ^ { 16 } = - 2$ 
B)  $4 ^ { 1 / 16 } = - 2$ 
C)  $4 ^ { - 2 } = \frac { 1 } { 16 }$ 
D)  $\left( \frac { 1 } { 16 } \right) ^ { - 2 } = 4$ 
E)  $4 ^ { - 2 } = - \frac { 1 } { 16 }$ 
A)  $4 ^ { 16 } = - 2$ 
B)  $4 ^ { 1 / 16 } = - 2$ 
C)  $4 ^ { - 2 } = \frac { 1 } { 16 }$ 
D)  $\left( \frac { 1 } { 16 } \right) ^ { - 2 } = 4$ 
E)  $4 ^ { - 2 } = - \frac { 1 } { 16 }$

Question 18

Identify the graph that represents the function.

Accepted Answer

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

Question 19

Use the regression feature of a graphing utility to find a logarithmic model $y = a + b \ln x$ for the data.
$( 1,3 ) , ( 2,4 ) , ( 3,4.5 ) , ( 4,5 ) , ( 5,5.1 ) , ( 6,5.2 ) , ( 7,5.5 )$

Accepted Answer

A)  $y = 3.030 + 1.470 \ln x$ 
B)  $y = 3.022 + 1.758 \ln x$ 
C)  $y = 3.115 + 0.842 \ln x$ 
D)  $y = 3.083 + 1.257 \ln x$ 
E)  $y = 3.167 + 0.555 \ln x$ 
A)  $y = 3.030 + 1.470 \ln x$ 
B)  $y = 3.022 + 1.758 \ln x$ 
C)  $y = 3.115 + 0.842 \ln x$ 
D)  $y = 3.083 + 1.257 \ln x$ 
E)  $y = 3.167 + 0.555 \ln x$

Question 20

Evaluate the logarithm $\log _ { 1 / 3 } 0.211$ using the change of base formula. Round to 3 decimal places.

Accepted Answer

A)  $- 1.556$ 
B)  $1.709$ 
C)  $0.706$ 
D)  $1.416$ 
E)  $- 0.676$ 
A)  $- 1.556$ 
B)  $1.709$ 
C)  $0.706$ 
D)  $1.416$ 
E)  $- 0.676$

Identify the vertical asymptote of the function $f ( x ) = 2 + \log ( x + 3 )$ .

Simplify the expression below.

Determine which $x$ -value below is a solution of the equation $\log _ { 7 } \left( \frac { 5 } { 3 } x \right) = 3$ . $x = \frac { 21 } { 125 } , x = \frac { 1029 } { 5 } , x = \frac { 189 } { 5 } , x = 21 , x = 15$

Identify the $x$ -intercept of the function $y = 3 + \log _ { 4 } x$ .

Evaluate the logarithm $\log _ { 1 / 3 } 0.113$ using the change of base formula. Round to 3 decimal places.

Solve the equation below algebraically. Round your result to three decimal places. $\frac { 1 + 2 \ln x } { x ^ { 5 } } = 0$

Solve the equation below algebraically. $- 2 x ^ { 2 } e ^ { 4 x } - 16 x e ^ { 4 x } = 0$

Solve the exponential equation below algebraically. Round your result to three decimal places.

Identify the graph of the function. $f ( x ) = \left( \frac { 1 } { 2 } \right) ^ { x }$

What is the value of the function $f ( x ) = 3.8 e ^ { - 1.5 x }$ at $x = 2.5 ?$ Round to 3 decimal places.

Condense the expression below to the logarithm of a single quantity. $\frac { 3 } { 2 } \log _ { 7 } ( z + 5 )$

Solve $\ln x ^ { 2 } = 3$ for $x$ .

What is the value of the function $f ( x ) = 3.3 e ^ { - 1.8 x }$ at $x = 2.5 ?$ Round to 3 decimal places.

Find the exponential model $y = a e ^ { b x }$ that fits the points shown in the table below. Round parameters to the nearest thousandth. x 0 1 y 5 10

Find the exact value of the logarithm without using a calculator, if possible. $\ln \frac { 1 } { \sqrt [ 2 ] { e } }$

Solve the logarithmic equation below algebraically. $\log _ { 6 } ( 2 x + 4 ) = \log _ { 6 } ( 5 x + 1 )$

Rewrite the logarithmic equation $\log _ { 4 } \frac { 1 } { 16 } = - 2$ in exponential form.

Identify the graph that represents the function.

Use the regression feature of a graphing utility to find a logarithmic model $y = a + b \ln x$ for the data. $( 1,3 ) , ( 2,4 ) , ( 3,4.5 ) , ( 4,5 ) , ( 5,5.1 ) , ( 6,5.2 ) , ( 7,5.5 )$

Evaluate the logarithm $\log _ { 1 / 3 } 0.211$ using the change of base formula. Round to 3 decimal places.

Functions and Their Graphs

Polynomial and Rational Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

Analytic Geometry in Three Dimensions

Calculus Practice Problems

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Exam 3: Exponential and Logarithmic Functions

Identify the vertical asymptote of the function f(x)=2+log⁡(x+3)f ( x ) = 2 + \log ( x + 3 )f(x)=2+log(x+3) .

Simplify the expression below.

Identify the xxx -intercept of the function y=3+log⁡4xy = 3 + \log _ { 4 } xy=3+log4​x .

Evaluate the logarithm log⁡1/30.113\log _ { 1 / 3 } 0.113log1/3​0.113 using the change of base formula. Round to 3 decimal places.

Solve the equation below algebraically. Round your result to three decimal places. 1+2ln⁡xx5=0\frac { 1 + 2 \ln x } { x ^ { 5 } } = 0x51+2lnx​=0

Solve the equation below algebraically. −2x2e4x−16xe4x=0- 2 x ^ { 2 } e ^ { 4 x } - 16 x e ^ { 4 x } = 0−2x2e4x−16xe4x=0

Solve the exponential equation below algebraically. Round your result to three decimal places.

Identify the graph of the function. f(x)=(12)xf ( x ) = \left( \frac { 1 } { 2 } \right) ^ { x }f(x)=(21​)x

What is the value of the function f(x)=3.8e−1.5xf ( x ) = 3.8 e ^ { - 1.5 x }f(x)=3.8e−1.5x at x=2.5?x = 2.5 ?x=2.5? Round to 3 decimal places.

Condense the expression below to the logarithm of a single quantity. 32log⁡7(z+5)\frac { 3 } { 2 } \log _ { 7 } ( z + 5 )23​log7​(z+5)

Solve ln⁡x2=3\ln x ^ { 2 } = 3lnx2=3 for xxx .

What is the value of the function f(x)=3.3e−1.8xf ( x ) = 3.3 e ^ { - 1.8 x }f(x)=3.3e−1.8x at x=2.5?x = 2.5 ?x=2.5? Round to 3 decimal places.

Find the exponential model y=aebxy = a e ^ { b x }y=aebx that fits the points shown in the table below. Round parameters to the nearest thousandth. x 0 1 y 5 10

Find the exact value of the logarithm without using a calculator, if possible. ln⁡1e2\ln \frac { 1 } { \sqrt [ 2 ] { e } }ln2e​1​

Solve the logarithmic equation below algebraically. log⁡6(2x+4)=log⁡6(5x+1)\log _ { 6 } ( 2 x + 4 ) = \log _ { 6 } ( 5 x + 1 )log6​(2x+4)=log6​(5x+1)

Rewrite the logarithmic equation log⁡4116=−2\log _ { 4 } \frac { 1 } { 16 } = - 2log4​161​=−2 in exponential form.

Identify the graph that represents the function.

Use the regression feature of a graphing utility to find a logarithmic model y=a+bln⁡xy = a + b \ln xy=a+blnx for the data. (1,3),(2,4),(3,4.5),(4,5),(5,5.1),(6,5.2),(7,5.5)( 1,3 ) , ( 2,4 ) , ( 3,4.5 ) , ( 4,5 ) , ( 5,5.1 ) , ( 6,5.2 ) , ( 7,5.5 )(1,3),(2,4),(3,4.5),(4,5),(5,5.1),(6,5.2),(7,5.5)

Evaluate the logarithm log⁡1/30.211\log _ { 1 / 3 } 0.211log1/3​0.211 using the change of base formula. Round to 3 decimal places.

Functions and Their Graphs

Polynomial and Rational Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

Analytic Geometry in Three Dimensions

Calculus Practice Problems

Filters

Identify the vertical asymptote of the function $f ( x ) = 2 + \log ( x + 3 )$ .

Identify the $x$ -intercept of the function $y = 3 + \log _ { 4 } x$ .

Evaluate the logarithm $\log _ { 1 / 3 } 0.113$ using the change of base formula. Round to 3 decimal places.

Solve the equation below algebraically. Round your result to three decimal places. $\frac { 1 + 2 \ln x } { x ^ { 5 } } = 0$

Solve the equation below algebraically. $- 2 x ^ { 2 } e ^ { 4 x } - 16 x e ^ { 4 x } = 0$

Identify the graph of the function. $f ( x ) = \left( \frac { 1 } { 2 } \right) ^ { x }$

What is the value of the function $f ( x ) = 3.8 e ^ { - 1.5 x }$ at $x = 2.5 ?$ Round to 3 decimal places.

Condense the expression below to the logarithm of a single quantity. $\frac { 3 } { 2 } \log _ { 7 } ( z + 5 )$

Solve $\ln x ^ { 2 } = 3$ for $x$ .

What is the value of the function $f ( x ) = 3.3 e ^ { - 1.8 x }$ at $x = 2.5 ?$ Round to 3 decimal places.

Find the exponential model $y = a e ^ { b x }$ that fits the points shown in the table below. Round parameters to the nearest thousandth. x 0 1 y 5 10

Find the exact value of the logarithm without using a calculator, if possible. $\ln \frac { 1 } { \sqrt [ 2 ] { e } }$

Solve the logarithmic equation below algebraically. $\log _ { 6 } ( 2 x + 4 ) = \log _ { 6 } ( 5 x + 1 )$

Rewrite the logarithmic equation $\log _ { 4 } \frac { 1 } { 16 } = - 2$ in exponential form.

Use the regression feature of a graphing utility to find a logarithmic model $y = a + b \ln x$ for the data. $( 1,3 ) , ( 2,4 ) , ( 3,4.5 ) , ( 4,5 ) , ( 5,5.1 ) , ( 6,5.2 ) , ( 7,5.5 )$

Evaluate the logarithm $\log _ { 1 / 3 } 0.211$ using the change of base formula. Round to 3 decimal places.