Question 1

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

Accepted Answer

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

Question 2

Solve the logarithmic equation below algebraically. Round your result to three decimal places. $4 \log _ { 3 } ( x - 4 ) = 13$

Accepted Answer

A)  41.280 
B)  43.869 
C)  43.623 
D)  35.658 
E)  39.534 
A)  41.280 
B)  43.869 
C)  43.623 
D)  35.658 
E)  39.534

Question 3

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 4

Solve the equation f ()xg= ()x algebraically. $\begin{array} { l } 
f ( x ) = \ln e ^ { 2 x + 2 } \
g ( x ) = 3 x + 6
\end{array}$

Accepted Answer

A)  5 
B)  4 
C)  -4 
D)  -5 
E)  2 
A)  5 
B)  4 
C)  -4 
D)  -5 
E)  2

Question 5

Approximate the solution of $9 e ^ { x + 3 } = 18$ to 3 decimal places. (You may use a graphing utility.)

Accepted Answer

A)  $- 0.803$ 
B)  $4.389$ 
C)  $- 2.699$ 
D)  $3.693$ 
E)  $- 2.307$ 
A)  $- 0.803$ 
B)  $4.389$ 
C)  $- 2.699$ 
D)  $3.693$ 
E)  $- 2.307$

Question 6

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

Graph IV:

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 7

What is the value of the function $f ( x ) = 3.3 e ^ { - 1.8 x } \text { at } x = 2.5 ?$ 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 8

Solve the logarithmic equation below algebraically. Round your result to three decimal places. $\ln \sqrt { x - 4 } = 2$

Accepted Answer

A)  38.489 
B)  28.421 
C)  3.668 
D)  74.354 
E)  58.598 
A)  38.489 
B)  28.421 
C)  3.668 
D)  74.354 
E)  58.598

Question 9

Evaluate the function $f ( x ) = \log _ { 2 } x$ at $x = \frac { 1 } { 4 }$ without using a calculator.

Accepted Answer

A)  $- 1$ 
B)  $- 2$ 
C)  $- 3$ 
D)  4 
E)  $\frac { 1 } { 4 }$ 
A)  $- 1$ 
B)  $- 2$ 
C)  $- 3$ 
D)  4 
E)  $\frac { 1 } { 4 }$

Question 10

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 11

Simplify the expression below. $6 + e ^ { \ln x ^ { 4 } }$

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)  $6 + 4 \ln x$

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.) $\ln \frac { x } { \sqrt [ 3 ] { x ^ { 2 } - 3 } }$

Accepted Answer

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

Question 13

A conservation organization releases 100 animals of an endangered species into a game preserve. The organization believes that the preserve has a carrying capacity of 900 Animals and that the growth of the herd will follow the logistic curve  $p ( t ) = \frac { 900 } { 1 + 10 e ^ { - 0.1656 t } }$ where t is measured in months. What is the population after 8 Months? Round your answer to the nearest animal.

Accepted Answer

A)  23 animals 
B)  243 animals 
C)  159 animals 
D)  308 animals 
E)  246 animals 
A)  23 animals 
B)  243 animals 
C)  159 animals 
D)  308 animals 
E)  246 animals

Question 14

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 15

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

Question 16

Evaluate the logarithm $\log _ { 7 } 126$ using the change of base formula. Round to 3 decimal places.

Accepted Answer

A)  4.836 
B)  0.402 
C)  2.485 
D)  9.411 
E)  2.100 
A)  4.836 
B)  0.402 
C)  2.485 
D)  9.411 
E)  2.100

Question 17

Solve the exponential equation below algebraically. $e ^ { - 6 x } = e ^ { x ^ { 2 } + 5 }$ 5

Accepted Answer

A)  -6 and -1 
B)  -5 and -1 
C)  -6 and -5 
D)  -5 and 6 
E)  -1 and 6 
A)  -6 and -1 
B)  -5 and -1 
C)  -6 and -5 
D)  -5 and 6 
E)  -1 and 6

Question 18

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 19

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

Question 20

Find the exact value of $\log _ { 8 } \sqrt [ 3 ] { 64 }$ without using a calculator.

Accepted Answer

A)  $\frac { 64 } { 3 }$ 
B)  $\frac { 3 } { 64 }$ 
C)  $\frac { 16 } { 3 }$ 
D)  $\frac { 2 } { 3 }$ 
E)  $- 1$ 
A)  $\frac { 64 } { 3 }$ 
B)  $\frac { 3 } { 64 }$ 
C)  $\frac { 16 } { 3 }$ 
D)  $\frac { 2 } { 3 }$ 
E)  $- 1$

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

Solve the logarithmic equation below algebraically. Round your result to three decimal places. $4 \log _ { 3 } ( x - 4 ) = 13$

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

Solve the equation f ()xg= ()x algebraically. f(x)= g(x)=3x+6

Approximate the solution of $9 e ^ { x + 3 } = 18$ to 3 decimal places. (You may use a graphing utility.)

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

Solve the logarithmic equation below algebraically. Round your result to three decimal places. $\ln \sqrt { x - 4 } = 2$

Evaluate the function $f ( x ) = \log _ { 2 } x$ at $x = \frac { 1 } { 4 }$ without using a calculator.

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

Simplify the expression below. $6 + e ^ { \ln x ^ { 4 } }$

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

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

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.

Evaluate the logarithm $\log _ { 7 } 126$ using the change of base formula. Round to 3 decimal places.

Solve the exponential equation below algebraically. $e ^ { - 6 x } = e ^ { x ^ { 2 } + 5 }$ 5

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

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.

Find the exact value of $\log _ { 8 } \sqrt [ 3 ] { 64 }$ without using a calculator.

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

Identify the graph of the function. f(x)=(12)x2f ( x ) = \left( \frac { 1 } { 2 } \right) ^ { x ^ { 2 } }f(x)=(21​)x2

Solve the logarithmic equation below algebraically. Round your result to three decimal places. 4log⁡3(x−4)=134 \log _ { 3 } ( x - 4 ) = 134log3​(x−4)=13

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)

Solve the equation f ()xg= ()x algebraically. f(x)= g(x)=3x+6

Approximate the solution of 9ex+3=189 e ^ { x + 3 } = 189ex+3=18 to 3 decimal places. (You may use a graphing utility.)

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

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

Solve the logarithmic equation below algebraically. Round your result to three decimal places. ln⁡x−4=2\ln \sqrt { x - 4 } = 2lnx−4​=2

Evaluate the function f(x)=log⁡2xf ( x ) = \log _ { 2 } xf(x)=log2​x at x=14x = \frac { 1 } { 4 }x=41​ without using a calculator.

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

Simplify the expression below. 6+eln⁡x46 + e ^ { \ln x ^ { 4 } }6+elnx4

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⁡xx2−33\ln \frac { x } { \sqrt [ 3 ] { x ^ { 2 } - 3 } }ln3x2−3​x​

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.

Evaluate the logarithm log⁡7126\log _ { 7 } 126log7​126 using the change of base formula. Round to 3 decimal places.

Solve the exponential equation below algebraically. e−6x=ex2+5e ^ { - 6 x } = e ^ { x ^ { 2 } + 5 }e−6x=ex2+5 5

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

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.

Find the exact value of log⁡8643\log _ { 8 } \sqrt [ 3 ] { 64 }log8​364​ without using a calculator.

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

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

Solve the logarithmic equation below algebraically. Round your result to three decimal places. $4 \log _ { 3 } ( x - 4 ) = 13$

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

Approximate the solution of $9 e ^ { x + 3 } = 18$ to 3 decimal places. (You may use a graphing utility.)

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

Solve the logarithmic equation below algebraically. Round your result to three decimal places. $\ln \sqrt { x - 4 } = 2$

Evaluate the function $f ( x ) = \log _ { 2 } x$ at $x = \frac { 1 } { 4 }$ without using a calculator.

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

Simplify the expression below. $6 + e ^ { \ln x ^ { 4 } }$

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

Evaluate the logarithm $\log _ { 7 } 126$ using the change of base formula. Round to 3 decimal places.

Solve the exponential equation below algebraically. $e ^ { - 6 x } = e ^ { x ^ { 2 } + 5 }$ 5

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

Find the exact value of $\log _ { 8 } \sqrt [ 3 ] { 64 }$ without using a calculator.