Question 1

Solve the equation below algebraically.
$- x ^ { 2 } e ^ { 2 x } - 7 x e ^ { 2 x } = 0$

Accepted Answer

A)  $- 7$ and $- 6$ 
B)  $- 6$ and $6$ 
C)  $- 7$ and $6$ 
D)  $0$ and $6$ 
E)  $- 7$ and $0$ 
A)  $- 7$ and $- 6$ 
B)  $- 6$ and $6$ 
C)  $- 7$ and $6$ 
D)  $0$ and $6$ 
E)  $- 7$ and $0$

Question 2

Find the exact value of $\log _ { 5 } \sqrt [ 3 ] { 25 }$ without using a calculator.

Accepted Answer

A)  $\frac { 25 } { 3 }$ 
B)  $\frac { 3 } { 25 }$ 
C)  $\frac { 10 } { 3 }$ 
D)  $\frac { 2 } { 3 }$ 
E)  $- 1$ 
A)  $\frac { 25 } { 3 }$ 
B)  $\frac { 3 } { 25 }$ 
C)  $\frac { 10 } { 3 }$ 
D)  $\frac { 2 } { 3 }$ 
E)  $- 1$

Question 3

Solve the equation $f ( x ) = g ( x )$ algebraically.
$\begin{array} { c } 
f ( x ) = \ln e ^ { 3 x - 9 } \
g ( x ) = x - 5
\end{array}$

Accepted Answer

A)  0 
B)  $- 1$ 
C)  2 
D)  5 
E)  1 
A)  0 
B)  $- 1$ 
C)  2 
D)  5 
E)  1

Question 4

Solve the logarithmic equation below algebraically.
$\ln ( x - 5 ) = \ln ( x + 1 ) - \ln ( x - 9 )$

Accepted Answer

A)  $- 12$ and $- 9$ 
B)  $4$ and $11$ 
C)  $- 12$ and 11 
D)  $- 9$ and 11 
E)  $- 12$ and 4 
A)  $- 12$ and $- 9$ 
B)  $4$ and $11$ 
C)  $- 12$ and 11 
D)  $- 9$ and 11 
E)  $- 12$ and 4

Question 5

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 6

Find the domain of the function below.
$$f ( x ) = \ln \left( \frac { x - 4 } { x + 5 } \right)$$

Accepted Answer

A)  $( - \infty , 4 )$ 
B)  $( - \infty , - 5 )$ 
C)  $( - 5 , \infty )$ 
D)  $( 4 , \infty )$ 
E)  $( - \infty , - 5 ) \cup ( 4 , \infty )$ 
A)  $( - \infty , 4 )$ 
B)  $( - \infty , - 5 )$ 
C)  $( - 5 , \infty )$ 
D)  $( 4 , \infty )$ 
E)  $( - \infty , - 5 ) \cup ( 4 , \infty )$

Question 7

Determine whether the scatter plot below could best be modeled by a linear model, a quadratic model, an exponential model, a logarithmic model, or a logistic model.

Accepted Answer

A)  a logistic model 
B)  an exponential model 
C)  a linear model 
D)  a logarithmic model 
E)  a quadratic model 
A)  a logistic model 
B)  an exponential model 
C)  a linear model 
D)  a logarithmic model 
E)  a quadratic model

Question 8

The sales S (in thousands of units) of a cleaning solution after x hundred dollars is spent on advertising are given by $S = 20 \left( 1 - e ^ { k x } \right)$ When $450 is spent on advertising, 2500 Units are sold. Complete the model by solving for k and use the model to estimate the Number of units that will be sold if advertising expenditures are raised to $650. Round Your answer to the nearest unit.

Accepted Answer

A)  3508 units 
B)  6401 units 
C)  2160 units 
D)  5663 units 
E)  8787 units 
A)  3508 units 
B)  6401 units 
C)  2160 units 
D)  5663 units 
E)  8787 units

Question 9

Rewrite the logarithm $\log _ { 4 } 25$ in terms of the natural logarithm.

Accepted Answer

A)  $\frac { \ln 25 } { \ln 4 }$ 
B)  $\frac { \ln 4 } { \ln 25 }$ 
C)  $\ln 4 \ln 25$ 
D)  $\frac { \ln 25 } { \log _ { 4 } e }$ 
E)  $\ln 25$ 
A)  $\frac { \ln 25 } { \ln 4 }$ 
B)  $\frac { \ln 4 } { \ln 25 }$ 
C)  $\ln 4 \ln 25$ 
D)  $\frac { \ln 25 } { \log _ { 4 } e }$ 
E)  $\ln 25$

Question 10

Identify the $x$-intercept of the function $f ( x ) = 3 \ln ( x - 4 )$.

Accepted Answer

A)  $x = 4$ 
B)  $x = 0$ 
C)  $x = 3$ 
D)  $x = 5$ 
E)  The function has no $x$-intercept. 
A)  $x = 4$ 
B)  $x = 0$ 
C)  $x = 3$ 
D)  $x = 5$ 
E)  The function has no $x$-intercept.

Question 11

Solve the exponential equation below algebraically.
$e ^ { - x ^ { 2 } } = e ^ { 2 x ^ { 2 } - 9 x }$

Accepted Answer

A)  0 and 3 
B)  $- 2$ and 0 
C)  $- 2$ and 3 
D)  $- 6$ and 3 
E)  $- 6$ and 0 
A)  0 and 3 
B)  $- 2$ and 0 
C)  $- 2$ and 3 
D)  $- 6$ and 3 
E)  $- 6$ and 0

Question 12

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) $\ln \sqrt [ 6 ] { t }$

Accepted Answer

A)  $\frac { 1 } { 6 } \ln t$ 
B)  $\ln t - \frac { 1 } { 6 }$ 
C)  $\ln t - 6$ 
D)  $\frac { 1 } { 3 } \ln t$ 
E)  $\ln t - \frac { 1 } { 3 }$ 
A)  $\frac { 1 } { 6 } \ln t$ 
B)  $\ln t - \frac { 1 } { 6 }$ 
C)  $\ln t - 6$ 
D)  $\frac { 1 } { 3 } \ln t$ 
E)  $\ln t - \frac { 1 } { 3 }$

Question 13

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

Accepted Answer

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

Question 14

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

Accepted Answer

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

Question 15

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 16

Condense the expression below to the logarithm of a single quantity.
$$5 [ \ln x - \ln ( x + 2 ) - \ln ( x - 2 ) ]$$

Accepted Answer

A)  $\quad \ln \left( x ^ { 5 } - \left( x ^ { 2 } - 4 \right) ^ { 5 } \right)$ 
B)  $\quad \ln \left( x ^ { 5 } - ( x + 2 ) ^ { 5 } - ( x - 2 ) ^ { 5 } \right)$ 
C)  $\ln \frac { x ^ { 5 } } { x ^ { 2 } - 4 }$ 
D)  $\ln \left( \left( \frac { x } { x ^ { 2 } - 4 } \right) ^ { 5 } \right)$ 
E)  $\quad \ln \frac { x } { \left( x ^ { 2 } - 4 \right) ^ { 5 } }$ 
A)  $\quad \ln \left( x ^ { 5 } - \left( x ^ { 2 } - 4 \right) ^ { 5 } \right)$ 
B)  $\quad \ln \left( x ^ { 5 } - ( x + 2 ) ^ { 5 } - ( x - 2 ) ^ { 5 } \right)$ 
C)  $\ln \frac { x ^ { 5 } } { x ^ { 2 } - 4 }$ 
D)  $\ln \left( \left( \frac { x } { x ^ { 2 } - 4 } \right) ^ { 5 } \right)$ 
E)  $\quad \ln \frac { x } { \left( x ^ { 2 } - 4 \right) ^ { 5 } }$

Question 17

Match the function $y = 3 + \ln ( x + 2 )$ with its graph.
Graph I:

Graph II:

Graph III:

Graph IV:
 
 Graph V:

Accepted Answer

A)  Graph II 
B)  Graph V 
C)  Graph III 
D)  Graph IV 
E)  Graph I 
A)  Graph II 
B)  Graph V 
C)  Graph III 
D)  Graph IV 
E)  Graph I

Question 18

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.)
$\ln \sqrt [ 4 ] { t }$

Accepted Answer

A)  $\frac { 1 } { 4 } \ln t$ 
B)  $\ln t - \frac { 1 } { 4 }$ 
C)  $\ln t - 4$ 
D)  $\frac { 1 } { 2 } \ln t$ 
E)  $\ln t - \frac { 1 } { 2 }$ 
A)  $\frac { 1 } { 4 } \ln t$ 
B)  $\ln t - \frac { 1 } { 4 }$ 
C)  $\ln t - 4$ 
D)  $\frac { 1 } { 2 } \ln t$ 
E)  $\ln t - \frac { 1 } { 2 }$

Question 19

Solve the equation below algebraically. Round your result to three decimal places.
$$\frac { 3 + \ln x } { x ^ { 2 } } = 0$$

Accepted Answer

A)  $- 1.876$ 
B)  $1.061$ 
C)  $- 1.390$ 
D)  $- 2.512$ 
E)  $0.050$ 
A)  $- 1.876$ 
B)  $1.061$ 
C)  $- 1.390$ 
D)  $- 2.512$ 
E)  $0.050$

Question 20

Solve the logarithmic equation below algebraically. Round your result to three decimal places. $\ln \left( x ^ { 2 } + 3 \right) = 4$

Accepted Answer

A)  $\pm 9.301$ 
B)  $\pm 6.772$ 
C)  $\pm 9.442$ 
D)  $\pm 4.318$ 
E)  $\pm 7.183$ 
A)  $\pm 9.301$ 
B)  $\pm 6.772$ 
C)  $\pm 9.442$ 
D)  $\pm 4.318$ 
E)  $\pm 7.183$

Solve the equation below algebraically. $- x ^ { 2 } e ^ { 2 x } - 7 x e ^ { 2 x } = 0$

Find the exact value of $\log _ { 5 } \sqrt [ 3 ] { 25 }$ without using a calculator.

Solve the equation $f ( x ) = g ( x )$ algebraically. f(x)= g(x)=x-5

Solve the logarithmic equation below algebraically. $\ln ( x - 5 ) = \ln ( x + 1 ) - \ln ( x - 9 )$

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

Find the domain of the function below. $f ( x ) = \ln \left( \frac { x - 4 } { x + 5 } \right)$

Determine whether the scatter plot below could best be modeled by a linear model, a quadratic model, an exponential model, a logarithmic model, or a logistic model.

Rewrite the logarithm $\log _ { 4 } 25$ in terms of the natural logarithm.

Identify the $x$ -intercept of the function $f ( x ) = 3 \ln ( x - 4 )$ .

Solve the exponential equation below algebraically. $e ^ { - x ^ { 2 } } = e ^ { 2 x ^ { 2 } - 9 x }$

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) $\ln \sqrt [ 6 ] { t }$

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

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

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

Condense the expression below to the logarithm of a single quantity. $5 [ \ln x - \ln ( x + 2 ) - \ln ( x - 2 ) ]$

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) $\ln \sqrt [ 4 ] { t }$

Solve the equation below algebraically. Round your result to three decimal places. $\frac { 3 + \ln x } { x ^ { 2 } } = 0$

Solve the logarithmic equation below algebraically. Round your result to three decimal places. $\ln \left( x ^ { 2 } + 3 \right) = 4$

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

Limits and an Introduction to Calculus

Review of Graphs, Equations, and Inequalities

Filters

Exam 3: Exponential and Logarithmic Functions

Solve the equation below algebraically. −x2e2x−7xe2x=0- x ^ { 2 } e ^ { 2 x } - 7 x e ^ { 2 x } = 0−x2e2x−7xe2x=0

Find the exact value of log⁡5253\log _ { 5 } \sqrt [ 3 ] { 25 }log5​325​ without using a calculator.

Solve the equation f(x)=g(x)f ( x ) = g ( x )f(x)=g(x) algebraically. f(x)= g(x)=x-5

Solve the logarithmic equation below algebraically. ln⁡(x−5)=ln⁡(x+1)−ln⁡(x−9)\ln ( x - 5 ) = \ln ( x + 1 ) - \ln ( x - 9 )ln(x−5)=ln(x+1)−ln(x−9)

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

Find the domain of the function below. f(x)=ln⁡(x−4x+5)f ( x ) = \ln \left( \frac { x - 4 } { x + 5 } \right)f(x)=ln(x+5x−4​)

Determine whether the scatter plot below could best be modeled by a linear model, a quadratic model, an exponential model, a logarithmic model, or a logistic model.

Rewrite the logarithm log⁡425\log _ { 4 } 25log4​25 in terms of the natural logarithm.

Identify the xxx -intercept of the function f(x)=3ln⁡(x−4)f ( x ) = 3 \ln ( x - 4 )f(x)=3ln(x−4) .

Solve the exponential equation below algebraically. e−x2=e2x2−9xe ^ { - x ^ { 2 } } = e ^ { 2 x ^ { 2 } - 9 x }e−x2=e2x2−9x

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⁡t6\ln \sqrt [ 6 ] { t }ln6t​

Simplify the expression log⁡3(127)3\log _ { 3 } \left( \frac { 1 } { 27 } \right) ^ { 3 }log3​(271​)3

Rewrite the logarithmic equation log⁡214=−2\log _ { 2 } \frac { 1 } { 4 } = - 2log2​41​=−2 in exponential form.

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

Condense the expression below to the logarithm of a single quantity. 5[ln⁡x−ln⁡(x+2)−ln⁡(x−2)]5 [ \ln x - \ln ( x + 2 ) - \ln ( x - 2 ) ]5[lnx−ln(x+2)−ln(x−2)]

Match the function y=3+ln⁡(x+2)y = 3 + \ln ( x + 2 )y=3+ln(x+2) with its graph. Graph I: Graph II: Graph III: Graph IV: Graph V:

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⁡t4\ln \sqrt [ 4 ] { t }ln4t​

Solve the equation below algebraically. Round your result to three decimal places. 3+ln⁡xx2=0\frac { 3 + \ln x } { x ^ { 2 } } = 0x23+lnx​=0

Solve the logarithmic equation below algebraically. Round your result to three decimal places. ln⁡(x2+3)=4\ln \left( x ^ { 2 } + 3 \right) = 4ln(x2+3)=4

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

Limits and an Introduction to Calculus

Review of Graphs, Equations, and Inequalities

Filters

Solve the equation below algebraically. $- x ^ { 2 } e ^ { 2 x } - 7 x e ^ { 2 x } = 0$

Find the exact value of $\log _ { 5 } \sqrt [ 3 ] { 25 }$ without using a calculator.

Solve the equation $f ( x ) = g ( x )$ algebraically. f(x)= g(x)=x-5

Solve the logarithmic equation below algebraically. $\ln ( x - 5 ) = \ln ( x + 1 ) - \ln ( x - 9 )$

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

Find the domain of the function below. $f ( x ) = \ln \left( \frac { x - 4 } { x + 5 } \right)$

Rewrite the logarithm $\log _ { 4 } 25$ in terms of the natural logarithm.

Identify the $x$ -intercept of the function $f ( x ) = 3 \ln ( x - 4 )$ .

Solve the exponential equation below algebraically. $e ^ { - x ^ { 2 } } = e ^ { 2 x ^ { 2 } - 9 x }$

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) $\ln \sqrt [ 6 ] { t }$

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

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

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

Condense the expression below to the logarithm of a single quantity. $5 [ \ln x - \ln ( x + 2 ) - \ln ( x - 2 ) ]$

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) $\ln \sqrt [ 4 ] { t }$

Solve the equation below algebraically. Round your result to three decimal places. $\frac { 3 + \ln x } { x ^ { 2 } } = 0$

Solve the logarithmic equation below algebraically. Round your result to three decimal places. $\ln \left( x ^ { 2 } + 3 \right) = 4$