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.

Question 2

A sample contains 75 grams of carbon $\left( { } ^ { 14 } C \right) . { } ^ { 14 } C$ has a half-life of 5715 years. How much ${ } ^ { 14 } C$ remains after 100 years? Round your answer to three decimal places.

Accepted Answer

A)  73.688 grams 
B)  74.096 grams 
C)  49.524 grams 
D)  1.312 grams 
E)  37.524 grams 
A)  73.688 grams 
B)  74.096 grams 
C)  49.524 grams 
D)  1.312 grams 
E)  37.524 grams

Question 3

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 4

Find the domain of the function below.
$$f ( x ) = \ln \left( \frac { x + 3 } { x - 5 } \right)$$

Accepted Answer

A)  $( - \infty , 5 )$ 
B)  $( - \infty , - 3 )$ 
C)  $( - 3 , \infty )$ 
D)  $( 5 , \infty )$ 
E)  $( - \infty , - 3 ) \cup ( 5 , \infty )$ 
A)  $( - \infty , 5 )$ 
B)  $( - \infty , - 3 )$ 
C)  $( - 3 , \infty )$ 
D)  $( 5 , \infty )$ 
E)  $( - \infty , - 3 ) \cup ( 5 , \infty )$

Question 5

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 6

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 7

Determine whether or not $x = \frac { 1 } { 3 } \left( e ^ { - 3 } + 1 \right) \text { is a solution to } \ln ( 3 x - 1 ) = - 3 \text {. }$

Accepted Answer

A) True 
 B)False

Question 8

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 9

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 10

Solve $\ln x ^ { 2 } = 13$ for $x$.

Accepted Answer

A)  $e ^ { 169 }$ 
B)  $10 ^ { \sqrt { 3 } }$ 
C)  $e ^ { \sqrt { 13 } }$ 
D)  $- e ^ { 13 / 2 } , e ^ { 13 / 2 }$ 
E)  no solution 
A)  $e ^ { 169 }$ 
B)  $10 ^ { \sqrt { 3 } }$ 
C)  $e ^ { \sqrt { 13 } }$ 
D)  $- e ^ { 13 / 2 } , e ^ { 13 / 2 }$ 
E)  no solution

Question 11

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 12

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 13

Solve for $x : 5 ^ { - x / 2 } = 0.0052$. Round to 3 decimal places.

Accepted Answer

A)  $6.535$ 
B)  $10.518$ 
C)  $13.737$ 
D)  $- 13.737$ 
E)  $- 3.268$ 
A)  $6.535$ 
B)  $10.518$ 
C)  $13.737$ 
D)  $- 13.737$ 
E)  $- 3.268$

Question 14

Condense the expression $\log _ { 5 } x + \log _ { 5 } 7$ to the logarithm of a single term.

Accepted Answer

A)  $\log ( 7 x ) ^ { 5 }$ 
B)  $\log _ { 5 } 7 x$ 
C)  $\log _ { 5 } 7 ^ { x }$ 
D)  $\log _ { 5 } x ^ { 7 }$ 
E)  $\log _ { 5 } ( x + 7 )$ 
A)  $\log ( 7 x ) ^ { 5 }$ 
B)  $\log _ { 5 } 7 x$ 
C)  $\log _ { 5 } 7 ^ { x }$ 
D)  $\log _ { 5 } x ^ { 7 }$ 
E)  $\log _ { 5 } ( x + 7 )$

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

Question 16

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 17

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 18

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 19

Solve the logarithmic equation below algebraically. Round your result to three decimal places.
$\log _ { 9 } 4 x - \log _ { 9 } ( 1 + \sqrt { x } ) = 2$

Accepted Answer

A)  $449.651$ 
B)  $618.792$ 
C)  $849.024$ 
D)  $5.103$ 
E)  $161.208$ 
A)  $449.651$ 
B)  $618.792$ 
C)  $849.024$ 
D)  $5.103$ 
E)  $161.208$

Question 20

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

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

A sample contains 75 grams of carbon $\left( { } ^ { 14 } C \right) . { } ^ { 14 } C$ has a half-life of 5715 years. How much ${ } ^ { 14 } C$ remains after 100 years? Round your answer to three decimal places.

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

Find the domain of the function below. $f ( x ) = \ln \left( \frac { x + 3 } { x - 5 } \right)$

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

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

Determine whether or not $x = \frac { 1 } { 3 } \left( e ^ { - 3 } + 1 \right) \text { is a solution to } \ln ( 3 x - 1 ) = - 3 \text {. }$

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

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

Solve $\ln x ^ { 2 } = 13$ for $x$ .

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

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

Solve for $x : 5 ^ { - x / 2 } = 0.0052$ . Round to 3 decimal places.

Condense the expression $\log _ { 5 } x + \log _ { 5 } 7$ to the logarithm of a single term.

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

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

Solve the logarithmic equation below algebraically. Round your result to three decimal places. $\log _ { 9 } 4 x - \log _ { 9 } ( 1 + \sqrt { x } ) = 2$

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

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

A sample contains 75 grams of carbon (14C).14C\left( { } ^ { 14 } C \right) . { } ^ { 14 } C(14C).14C has a half-life of 5715 years. How much 14C{ } ^ { 14 } C14C remains after 100 years? Round your answer to three decimal places.

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.

Find the domain of the function below. f(x)=ln⁡(x+3x−5)f ( x ) = \ln \left( \frac { x + 3 } { x - 5 } \right)f(x)=ln(x−5x+3​)

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

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) )

Determine whether or not x=13(e−3+1) is a solution to ln⁡(3x−1)=−3. x = \frac { 1 } { 3 } \left( e ^ { - 3 } + 1 \right) \text { is a solution to } \ln ( 3 x - 1 ) = - 3 \text {. }x=31​(e−3+1) is a solution to ln(3x−1)=−3.

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

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

Solve ln⁡x2=13\ln x ^ { 2 } = 13lnx2=13 for xxx .

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

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

Solve for x:5−x/2=0.0052x : 5 ^ { - x / 2 } = 0.0052x:5−x/2=0.0052 . Round to 3 decimal places.

Condense the expression log⁡5x+log⁡57\log _ { 5 } x + \log _ { 5 } 7log5​x+log5​7 to the logarithm of a single term.

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

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

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. Round your result to three decimal places. log⁡94x−log⁡9(1+x)=2\log _ { 9 } 4 x - \log _ { 9 } ( 1 + \sqrt { x } ) = 2log9​4x−log9​(1+x​)=2

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

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

A sample contains 75 grams of carbon $\left( { } ^ { 14 } C \right) . { } ^ { 14 } C$ has a half-life of 5715 years. How much ${ } ^ { 14 } C$ remains after 100 years? Round your answer to three decimal places.

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

Find the domain of the function below. $f ( x ) = \ln \left( \frac { x + 3 } { x - 5 } \right)$

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

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

Determine whether or not $x = \frac { 1 } { 3 } \left( e ^ { - 3 } + 1 \right) \text { is a solution to } \ln ( 3 x - 1 ) = - 3 \text {. }$

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

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

Solve $\ln x ^ { 2 } = 13$ for $x$ .

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

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

Solve for $x : 5 ^ { - x / 2 } = 0.0052$ . Round to 3 decimal places.

Condense the expression $\log _ { 5 } x + \log _ { 5 } 7$ to the logarithm of a single term.

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

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

Solve the logarithmic equation below algebraically. Round your result to three decimal places. $\log _ { 9 } 4 x - \log _ { 9 } ( 1 + \sqrt { x } ) = 2$

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