Question 1

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 2

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 3

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)  $\pm 7.183$

Question 4

Simplify the expression $\log _ { 5 } 150$.

Accepted Answer

A)  $2 + \log _ { 5 } 6$ 
B)  $30 \log _ { 5 } 2$ 
C)  6 
D)  $2 \log _ { 5 } 6$ 
E)  The expression cannot be simplified. 
A)  $2 + \log _ { 5 } 6$ 
B)  $30 \log _ { 5 } 2$ 
C)  6 
D)  $2 \log _ { 5 } 6$ 
E)  The expression cannot be simplified.

Question 5

Solve the equation below algebraically. Round your result to three decimal places.
$$2 x \ln \left( \frac { 1 } { x } \right) - x = 0$$

Accepted Answer

A)  $0.607$ 
B)  $0.000$ 
C)  $- 2.009$ 
D)  $2.086$ 
E)  $- 1.058$ 
A)  $0.607$ 
B)  $0.000$ 
C)  $- 2.009$ 
D)  $2.086$ 
E)  $- 1.058$

Question 6

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 7

Find the exponential model $y = a e ^ { b x }$ that fits the points shown in the graph when $A = ( 0,3 )$ and $B = ( - 3,24 )$. Round parameters to the nearest thousandth.

Accepted Answer

A)  $y = - 3 e ^ { 0.693 x }$ 
B)  $y = - 0.693 e ^ { - 3 x }$ 
C)  $y = 3 e ^ { - 0.693 x }$ 
D)  $y = 3 e ^ { 0.693 x }$ 
E)  $y = - 0.693 e ^ { 3 x }$ 
A)  $y = - 3 e ^ { 0.693 x }$ 
B)  $y = - 0.693 e ^ { - 3 x }$ 
C)  $y = 3 e ^ { - 0.693 x }$ 
D)  $y = 3 e ^ { 0.693 x }$ 
E)  $y = - 0.693 e ^ { 3 x }$

Question 8

Solve the logarithmic equation below algebraically.
$$\ln ( x - 6 ) = \ln ( x + 6 ) - \ln ( x - 4 )$$

Accepted Answer

A)  $- 2$ and 5 
B)  2 and 9 
C)  5 and 9 
D)  $- 2$ and 9 
E)  2 and 5 
A)  $- 2$ and 5 
B)  2 and 9 
C)  5 and 9 
D)  $- 2$ and 9 
E)  2 and 5

Question 9

Evaluate the function $f ( x ) = \log _ { 4 } x$ at $x = \frac { 1 } { 16 }$ without using a calculator.

Accepted Answer

A)  $- 1$ 
B)  $- 2$ 
C)  $- 3$ 
D)  16 
E)  $\frac { 1 } { 16 }$ 
A)  $- 1$ 
B)  $- 2$ 
C)  $- 3$ 
D)  16 
E)  $\frac { 1 } { 16 }$

Question 10

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.)
$$\ln \frac { x } { \sqrt [ 3 ] { x ^ { 5 } - 2 } }$$

Accepted Answer

A)  $\quad \ln x + \frac { 5 } { 3 } \ln ( x - 2 )$ 
B)  $- \frac { 2 } { 3 } \ln x + \ln ( - 2 )$ 
C)  $- \frac { 2 } { 3 } \ln x - \ln ( - 2 )$ 
D)  $\ln x - \frac { 5 } { 3 } \ln ( x - 2 )$ 
E)  $\quad \ln x - \frac { 1 } { 3 } \ln \left( x ^ { 5 } - 2 \right)$ 
A)  $\quad \ln x + \frac { 5 } { 3 } \ln ( x - 2 )$ 
B)  $- \frac { 2 } { 3 } \ln x + \ln ( - 2 )$ 
C)  $- \frac { 2 } { 3 } \ln x - \ln ( - 2 )$ 
D)  $\ln x - \frac { 5 } { 3 } \ln ( x - 2 )$ 
E)  $\quad \ln x - \frac { 1 } { 3 } \ln \left( x ^ { 5 } - 2 \right)$

Question 11

D 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)  $x = 15$

Question 12

Solve the exponential equation below algebraically. Round your result to three decimal places.
$$\frac { 500 } { 1 + e ^ { - x } } = 150$$

Accepted Answer

A)  $0.794$ 
B)  $- 1.483$ 
C)  $- 3.655$ 
D)  $- 3.796$ 
E)  $- 0.847$ 
A)  $0.794$ 
B)  $- 1.483$ 
C)  $- 3.655$ 
D)  $- 3.796$ 
E)  $- 0.847$

Question 13

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 14

Match the function $y = 1 + \ln ( x + 4 )$ with its graph.
Graph I:
 
Graph II:
 
Graph III:
 
Graph IV:
 
 Graph V:

Accepted Answer

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

Question 15

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 16

Condense the expression below to the logarithm of a single quantity.
$$\frac { 5 } { 2 } \log _ { 8 } ( z - 1 )$$

Accepted Answer

A)  $\log _ { 8 } ( \sqrt [ 5 ] { z } - 1 )$ 
B)  $\log _ { 8 } \left( \sqrt { z ^ { 5 } } - 1 \right)$ 
C)  $\log _ { 8 } \sqrt [ 5 ] { z - 1 }$ 
D)  $\log _ { 8 } \frac { 5 ( z - 1 ) } { 2 }$ 
E)  $\log _ { 8 } \sqrt { ( z - 1 ) ^ { 5 } }$ 
A)  $\log _ { 8 } ( \sqrt [ 5 ] { z } - 1 )$ 
B)  $\log _ { 8 } \left( \sqrt { z ^ { 5 } } - 1 \right)$ 
C)  $\log _ { 8 } \sqrt [ 5 ] { z - 1 }$ 
D)  $\log _ { 8 } \frac { 5 ( z - 1 ) } { 2 }$ 
E)  $\log _ { 8 } \sqrt { ( z - 1 ) ^ { 5 } }$

Question 17

Find the domain of the function below.
$$f ( x ) = \ln \left( \frac { x } { x ^ { 2 } + 16 } \right)$$

Accepted Answer

A)  $( 4 , \infty )$ 
B)  $( - \infty , - 4 ) \cup ( 4 , \infty )$ 
C)  $( - \infty , 0 ) \cup ( 4 , \infty )$ 
D)  $( 0 , \infty )$ 
E)  $( - \infty , - 4 ) \cup ( 0 , \infty )$ 
A)  $( 4 , \infty )$ 
B)  $( - \infty , - 4 ) \cup ( 4 , \infty )$ 
C)  $( - \infty , 0 ) \cup ( 4 , \infty )$ 
D)  $( 0 , \infty )$ 
E)  $( - \infty , - 4 ) \cup ( 0 , \infty )$

Question 18

Find the exact value of the logarithm without using a calculator, if possible. $\log _ { 4 } 128 + \log _ { 4 } 8$

Accepted Answer

A)  1 
B)  2 
C)  3 
D)  4 
E)  not possible without using a calculator 
A)  1 
B)  2 
C)  3 
D)  4 
E)  not possible without using a calculator

Question 19

Solve the equation below for $x$. $\log _ { 5 } 5 ^ { 7 } = x$

Accepted Answer

A)  8 
B)  2 
C)  5 
D)  9 
E)  7 
A)  8 
B)  2 
C)  5 
D)  9 
E)  7

Question 20

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)  $\ln t - \frac { 1 } { 2 }$

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

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

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

Simplify the expression $\log _ { 5 } 150$ .

Solve the equation below algebraically. Round your result to three decimal places. $2 x \ln \left( \frac { 1 } { x } \right) - x = 0$

Solve the logarithmic equation below algebraically. $\ln ( x - 6 ) = \ln ( x + 6 ) - \ln ( x - 4 )$

Evaluate the function $f ( x ) = \log _ { 4 } x$ at $x = \frac { 1 } { 16 }$ without using a calculator.

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) $\ln \frac { x } { \sqrt [ 3 ] { x ^ { 5 } - 2 } }$

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

Solve the exponential equation below algebraically. Round your result to three decimal places. $\frac { 500 } { 1 + e ^ { - x } } = 150$

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

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

Condense the expression below to the logarithm of a single quantity. $\frac { 5 } { 2 } \log _ { 8 } ( z - 1 )$

Find the domain of the function below. $f ( x ) = \ln \left( \frac { x } { x ^ { 2 } + 16 } \right)$

Find the exact value of the logarithm without using a calculator, if possible. $\log _ { 4 } 128 + \log _ { 4 } 8$

Solve the equation below for $x$ . $\log _ { 5 } 5 ^ { 7 } = x$

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

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

Filters

Exam 3: Exponential and Logarithmic Functions

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

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 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

Simplify the expression log⁡5150\log _ { 5 } 150log5​150 .

Solve the equation below algebraically. Round your result to three decimal places. 2xln⁡(1x)−x=02 x \ln \left( \frac { 1 } { x } \right) - x = 02xln(x1​)−x=0

Find the exponential model y=aebxy = a e ^ { b x }y=aebx that fits the points shown in the graph when A=(0,3)A = ( 0,3 )A=(0,3) and B=(−3,24)B = ( - 3,24 )B=(−3,24) . Round parameters to the nearest thousandth.

Solve the logarithmic equation below algebraically. ln⁡(x−6)=ln⁡(x+6)−ln⁡(x−4)\ln ( x - 6 ) = \ln ( x + 6 ) - \ln ( x - 4 )ln(x−6)=ln(x+6)−ln(x−4)

Evaluate the function f(x)=log⁡4xf ( x ) = \log _ { 4 } xf(x)=log4​x at x=116x = \frac { 1 } { 16 }x=161​ without using a calculator.

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⁡xx5−23\ln \frac { x } { \sqrt [ 3 ] { x ^ { 5 } - 2 } }ln3x5−2​x​

Solve the exponential equation below algebraically. Round your result to three decimal places. 5001+e−x=150\frac { 500 } { 1 + e ^ { - x } } = 1501+e−x500​=150

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

Match the function y=1+ln⁡(x+4)y = 1 + \ln ( x + 4 )y=1+ln(x+4) with its graph. Graph I: Graph II: Graph III: Graph IV: Graph V:

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

Condense the expression below to the logarithm of a single quantity. 52log⁡8(z−1)\frac { 5 } { 2 } \log _ { 8 } ( z - 1 )25​log8​(z−1)

Find the domain of the function below. f(x)=ln⁡(xx2+16)f ( x ) = \ln \left( \frac { x } { x ^ { 2 } + 16 } \right)f(x)=ln(x2+16x​)

Find the exact value of the logarithm without using a calculator, if possible. log⁡4128+log⁡48\log _ { 4 } 128 + \log _ { 4 } 8log4​128+log4​8

Solve the equation below for xxx . log⁡557=x\log _ { 5 } 5 ^ { 7 } = xlog5​57=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⁡t4\ln \sqrt [ 4 ] { t }ln4t​

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

Filters

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

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

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

Simplify the expression $\log _ { 5 } 150$ .

Solve the equation below algebraically. Round your result to three decimal places. $2 x \ln \left( \frac { 1 } { x } \right) - x = 0$

Solve the logarithmic equation below algebraically. $\ln ( x - 6 ) = \ln ( x + 6 ) - \ln ( x - 4 )$

Evaluate the function $f ( x ) = \log _ { 4 } x$ at $x = \frac { 1 } { 16 }$ without using a calculator.

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) $\ln \frac { x } { \sqrt [ 3 ] { x ^ { 5 } - 2 } }$

Solve the exponential equation below algebraically. Round your result to three decimal places. $\frac { 500 } { 1 + e ^ { - x } } = 150$

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

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

Condense the expression below to the logarithm of a single quantity. $\frac { 5 } { 2 } \log _ { 8 } ( z - 1 )$

Find the domain of the function below. $f ( x ) = \ln \left( \frac { x } { x ^ { 2 } + 16 } \right)$

Find the exact value of the logarithm without using a calculator, if possible. $\log _ { 4 } 128 + \log _ { 4 } 8$

Solve the equation below for $x$ . $\log _ { 5 } 5 ^ { 7 } = x$

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