Question 1

Solve the problem.
-Use the graph of $y = 2 ^ { - x }$ to graph the function . Write the domain and range in interval notation $f ( x ) = 2 ^ { - x }$

Accepted Answer

A) 
 
Domain: $( - \infty , \infty ) \quad$ 
Range: $( 0 , \infty ) \quad$ 
B) 
 
Domain: $( - \infty , \infty )$
 Range: $( - \infty , 0 )$ 
C) 
 
Domain: $( - \infty , \infty )$
Range: $( - \infty , 0 )$ 
D) 
 
Domain: $( - \infty , \infty )$
Range: $( 0 , \infty )$ 
A) 
 
Domain: $( - \infty , \infty ) \quad$ 
Range: $( 0 , \infty ) \quad$ 
B) 
 
Domain: $( - \infty , \infty )$
 Range: $( - \infty , 0 )$ 
C) 
 
Domain: $( - \infty , \infty )$
Range: $( - \infty , 0 )$ 
D) 
 
Domain: $( - \infty , \infty )$
Range: $( 0 , \infty )$

Question 2

Solve the problem.
-The enrollment at one Midwestern college is approximated by the formula $P = 3000 ( 1.08 ) ^ { t }$ where $t$ is the number of years after 2000 . What is the first year in which you would expect enrollment to surpass 3,700 ?

Accepted Answer

A)  2,004 
B)  2,002 
C)  2,001 
D)  2,003 
A)  2,004 
B)  2,002 
C)  2,001 
D)  2,003

Question 3

Simplify the expression without using a calculator.
-$\log _ { 5 / 4 } \left( \frac { 256 } { 625 } \right)$

Accepted Answer

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

Question 4

Determine if the statement is true or false.
-$\log _ { 4 } \left( \frac { 1 } { p } \right) = - \log _ { 4 } p$

Accepted Answer

A) True 
 B)False

Question 5

Solve the problem.
-Use $\log _ { b } 3 \approx 0.319$, use the properties of logarithms to approximate the following. Do not use a calcu $\log _ { b } 81$

Accepted Answer

A)  $4.319$ 
B)  $1.276$ 
C)  $- 3.681$ 
D)  $0.08$ 
A)  $4.319$ 
B)  $1.276$ 
C)  $- 3.681$ 
D)  $0.08$

Question 6

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4
decimal places if necessary.
-$$\log _ { 5 } ( 6 x - 14 ) = 1 + \log _ { 5 } ( x - 2 )$$

Accepted Answer

A)  $\left\{ \frac { 17 } { 5 } \right\}$ 
B)  $\{ - 24 \}$ 
C)  $\{ 4 \}$ 
D)  $\left\{ \frac { 4 } { 11 } \right\}$ 
A)  $\left\{ \frac { 17 } { 5 } \right\}$ 
B)  $\{ - 24 \}$ 
C)  $\{ 4 \}$ 
D)  $\left\{ \frac { 4 } { 11 } \right\}$

Question 7

Choose the one alternative that best completes the statement or answers the question.
A relation in x and y is given. Determine if the relation defines y as a one-to-one function of x.
-$\{ ( - 6,8 ) , - 5 , - 8 ) , ( 4 , - 6 ) , ( 1,3 ) \}$

Accepted Answer

A)  Yes 
B)  No 
A)  Yes 
B)  No

Question 8

Solve the equation. Write the solution set with the exact values given in terms of natural or common
logarithms. Also give approximate solutions to 4 decimal places, if necessary.
-$5 ^ { 5 x + 6 } = 4 ^ { 8 x }$; use natural logarithms

Accepted Answer

A)  $\left\{ \frac { 6 \ln 5 } { 5 \ln 5 - 8 \ln 4 } \right\} ; x \approx - 3.1732$ 
B)  $\left\{ \frac { 6 \ln 5 } { 8 \ln 4 - 5 \ln 5 } \right\} ; x \approx 3.1732$ 
C)  $\left\{ \frac { \ln 6 } { \ln 8 - \ln 5 } \right\} ; x \approx 3.8122$ 
D)  \{ \} 
A)  $\left\{ \frac { 6 \ln 5 } { 5 \ln 5 - 8 \ln 4 } \right\} ; x \approx - 3.1732$ 
B)  $\left\{ \frac { 6 \ln 5 } { 8 \ln 4 - 5 \ln 5 } \right\} ; x \approx 3.1732$ 
C)  $\left\{ \frac { \ln 6 } { \ln 8 - \ln 5 } \right\} ; x \approx 3.8122$ 
D)  \{ \}

Question 9

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4
decimal places if necessary.
-$2 \log _ { 6 } ( 6 p - 8 ) - 4 = - 2$

Accepted Answer

A)  $\left\{ \frac { 7 } { 3 } \right\}$ 
B)  $\left\{ - \frac { 7 } { 6 } \right\}$ 
C)  $\left\{ \frac { 3 } { 2 } \right\}$ 
D)  $\left\{ - \frac { 1 } { 3 } \right\}$ 
A)  $\left\{ \frac { 7 } { 3 } \right\}$ 
B)  $\left\{ - \frac { 7 } { 6 } \right\}$ 
C)  $\left\{ \frac { 3 } { 2 } \right\}$ 
D)  $\left\{ - \frac { 1 } { 3 } \right\}$

Question 10

Use transformations of the graph $y = e ^ { x }$ to graph the function. Write the domain and range in interval
notation.
-$$f ( x ) = - e ^ { x } + 1$$

Accepted Answer

A) 
 
Domain: $ (-\infty, \infty) $
Range: $ (-\infty, 0) $ 
B) 
 
Domain: $ (-\infty, \infty) $
Range: $ (-\infty, 1) $ 
C) 
 
Domain: $( - \infty , \infty )$
Range: $( - \infty , - 1 )$ 
D) 
 
Domain: $( - \infty , \infty )$
Range: $( - \infty , 0 )$ 
A) 
 
Domain: $ (-\infty, \infty) $
Range: $ (-\infty, 0) $ 
B) 
 
Domain: $ (-\infty, \infty) $
Range: $ (-\infty, 1) $ 
C) 
 
Domain: $( - \infty , \infty )$
Range: $( - \infty , - 1 )$ 
D) 
 
Domain: $( - \infty , \infty )$
Range: $( - \infty , 0 )$

Question 11

Choose the one alternative that best completes the statement or answers the question.
Evaluate the function at the given value of x. Round to 4 decimal places if necessary.
-$h ( x ) = \left( \frac { 1 } { 3 } \right) ^ { x } ; \quad h ( - 3 )$

Accepted Answer

A)  $\frac { 1 } { 9 }$ 
B)  9 
C)  27 
D)  $\frac { 1 } { 27 }$ 
A)  $\frac { 1 } { 9 }$ 
B)  9 
C)  27 
D)  $\frac { 1 } { 27 }$

Question 12

Choose the one alternative that best completes the statement or answers the question.
Solve for the indicated variable.
-$A = P e ^ { r t }$ for $r$

Accepted Answer

A)  $r = \frac { \ln \left( \frac { A } { P } \right) } { \ln ( t ) }$ 
B)  $r = \frac { \ln ( A - P ) } { t }$ 
C) $r = \frac { \ln \left( \frac { A } { P } \right) } { t }$ 
D) $r = \frac { \ln \left( \frac { P } { A } \right) } { t }$ 
A)  $r = \frac { \ln \left( \frac { A } { P } \right) } { \ln ( t ) }$ 
B)  $r = \frac { \ln ( A - P ) } { t }$ 
C) $r = \frac { \ln \left( \frac { A } { P } \right) } { t }$ 
D) $r = \frac { \ln \left( \frac { P } { A } \right) } { t }$

Question 13

Write the equation in logarithmic form.
-$$10 ^ { 5 } = 100,000$$

Accepted Answer

A)  $\log _ { 5 } 100,000 = 10$ 
B)  $\log _ { 100,000 } 5 = 10$ 
C)  $100,000 = \log 5$ 
D)  $\log 100,000 = 5$ 
A)  $\log _ { 5 } 100,000 = 10$ 
B)  $\log _ { 100,000 } 5 = 10$ 
C)  $100,000 = \log 5$ 
D)  $\log 100,000 = 5$

Question 14

Solve the problem.
-Newton's law of cooling indicates that the temperature of a warm object will decrease exponentially with time and will approach the temperature of the surrounding air. The temperature $T ( t )$ is modeled by $T ( t ) = T _ { a } + \left( T _ { 0 } - T _ { a } \right) e ^ { - k t }$. In this model, $T _ { a }$ represents the temperature of the surrounding air, $T _ { 0 }$ represents the initial temperature of the object and $t$ is the time after the object starts cooling. The value of $k$ is the cooling rate and is a constant related to the physical properties of the object.
A cake comes out of the oven at $335 ^ { \circ } \mathrm { F }$ and is placed on a cooling rack in a $70 ^ { \circ } \mathrm { F }$ kitchen. After checking the temperature several minutes later, it is determined that the cooling rate $k$ is $0.050$. Write a function that models the temperature $T ( t )$ (in ${ } ^ { \circ } \mathrm { F }$ ) of the cake $t$ minutes after being removed from the oven.

Accepted Answer

A)  $T ( t ) = 335 + 70 e ^ { 0.050 t }$ 
B)  $T ( t ) = 70 + 265 e ^ { 0.050 t }$ 
C)  $T ( t ) = 70 + 265 e ^ { - 0.050 t }$ 
D)  $T ( t ) = 70 + 335 e ^ { 0.050 t }$ 
A)  $T ( t ) = 335 + 70 e ^ { 0.050 t }$ 
B)  $T ( t ) = 70 + 265 e ^ { 0.050 t }$ 
C)  $T ( t ) = 70 + 265 e ^ { - 0.050 t }$ 
D)  $T ( t ) = 70 + 335 e ^ { 0.050 t }$

Question 15

Approximate the value of the logarithm to four decimal places.
-$\log 512$

Accepted Answer

A)  2.8392 
B)  6.2383 
C)  2.6016 
D)  2.7093 
A)  2.8392 
B)  6.2383 
C)  2.6016 
D)  2.7093

Question 16

Write as the sum or difference of logarithms and fully simplify, if possible. Assume the variable
represents a positive real number.
-$\ln \sqrt [ 8 ] { \frac { e ^ { 7 } } { x ^ { 4 } + y } }$

Accepted Answer

A)  $\frac { 7 } { 8 } \ln e - \frac { 1 } { 8 } \ln \left( x ^ { 4 } + y \right)$ 
B)  $\frac { 7 } { 8 } - \frac { 4 } { 8 } \ln x + \frac { 1 } { 8 } \ln y$ 
C)  $\frac { 7 } { 8 } - \frac { 1 } { 8 } \ln \left( x ^ { 4 } + y \right)$ 
D)  $\frac { 7 } { 8 } + \frac { 1 } { 8 } \ln \left( x ^ { 4 } + y \right)$ 
A)  $\frac { 7 } { 8 } \ln e - \frac { 1 } { 8 } \ln \left( x ^ { 4 } + y \right)$ 
B)  $\frac { 7 } { 8 } - \frac { 4 } { 8 } \ln x + \frac { 1 } { 8 } \ln y$ 
C)  $\frac { 7 } { 8 } - \frac { 1 } { 8 } \ln \left( x ^ { 4 } + y \right)$ 
D)  $\frac { 7 } { 8 } + \frac { 1 } { 8 } \ln \left( x ^ { 4 } + y \right)$

Question 17

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4
decimal places if necessary.
-$\log \left( x ^ { 2 } - 2 x \right) = \log 15$

Accepted Answer

A)  $\{ 5 , - 3 \}$ 
B)  $\{ - 5,3 \}$ 
C)  $\{ 5 \}$ 
D) {  } 
A)  $\{ 5 , - 3 \}$ 
B)  $\{ - 5,3 \}$ 
C)  $\{ 5 \}$ 
D) {  }

Question 18

Solve the problem.
-A pie comes out of the oven at $275 ^ { \circ } \mathrm { F }$ and is placed to cool in a $70 ^ { \circ } \mathrm { F }$ kitchen. The temperature of the pie $T$ (in ${ } ^ { \circ } \mathrm { F }$ ) after $t$ minutes is given by $T = 70 + 205 e ^ { - 0.018 t }$. The pie is cool enough to cut when the temperature reaches $130 ^ { \circ } \mathrm { F }$. How long will this take? Round to the nearest minute.

Accepted Answer

A)  68 minutes 
B)  25 minutes 
C)  56 minutes 
D)  85 minutes 
A)  68 minutes 
B)  25 minutes 
C)  56 minutes 
D)  85 minutes

Question 19

Determine whether the two functions are inverses.
-$f ( x ) = 6 x + 6 \text { and } g ( x ) = \frac { 6 + x } { 6 }$

Accepted Answer

A)  No 
B)  Yes 
A)  No 
B)  Yes

Question 20

Solve the problem.
-Newton's law of cooling indicates that the temperature of a warm object will decrease exponentially with time and will approach the temperature of the surrounding air. The temperature $T ( t )$ is modeled by $T ( t ) = T _ { a } + \left( T _ { 0 } - T _ { a } \right) e ^ { - k t }$. In this model, $T _ { a }$ represents the temperature of the surrounding air, $T _ { 0 }$ represents the initial temperature of the object and $t$ is the time after the object starts cooling. The value of $k$ is the cooling rate and is a constant related to the physical properties of the object.
Water in a water heater is originally $131 ^ { \circ } \mathrm { F }$. The water heater is shut off and the water cools to the temperature of the surrounding air, which is $80 ^ { \circ } \mathrm { F }$. The water cools slowly because of the insulation inside the heater, and the rate of cooling is $0.00348$. Dominic does not like to shower with water less than $116 ^ { \circ } \mathrm { F }$. If Dominic waits $24 \mathrm { hr }$ after the heater is shut off, will the water still be warm enough for a shower? How warm will the water be?

Accepted Answer

A)  No; the temperature after $24 \mathrm { hr }$ will be approximately $80 ^ { \circ } \mathrm { F }$ 
B)  Yes; the temperature after $24 \mathrm { hr }$ will be approximately $128 ^ { \circ } \mathrm { F }$ 
C)  Yes; the temperature after $24 \mathrm { hr }$ will be approximately $127 ^ { \circ } \mathrm { F }$ 
D)  $\mathrm { No }$; the temperature after $24 \mathrm { hr }$ will be approximately $106 ^ { \circ } \mathrm { F }$ 
A)  No; the temperature after $24 \mathrm { hr }$ will be approximately $80 ^ { \circ } \mathrm { F }$ 
B)  Yes; the temperature after $24 \mathrm { hr }$ will be approximately $128 ^ { \circ } \mathrm { F }$ 
C)  Yes; the temperature after $24 \mathrm { hr }$ will be approximately $127 ^ { \circ } \mathrm { F }$ 
D)  $\mathrm { No }$; the temperature after $24 \mathrm { hr }$ will be approximately $106 ^ { \circ } \mathrm { F }$

Solve the problem. -Use the graph of $y = 2 ^ { - x }$ to graph the function . Write the domain and range in interval notation $f ( x ) = 2 ^ { - x }$

Solve the problem. -The enrollment at one Midwestern college is approximated by the formula $P = 3000 ( 1.08 ) ^ { t }$ where $t$ is the number of years after 2000 . What is the first year in which you would expect enrollment to surpass 3,700 ?

Simplify the expression without using a calculator. - $\log _ { 5 / 4 } \left( \frac { 256 } { 625 } \right)$

Determine if the statement is true or false. - $\log _ { 4 } \left( \frac { 1 } { p } \right) = - \log _ { 4 } p$

Solve the problem. -Use $\log _ { b } 3 \approx 0.319$ , use the properties of logarithms to approximate the following. Do not use a calcu $\log _ { b } 81$

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. - $\log _ { 5 } ( 6 x - 14 ) = 1 + \log _ { 5 } ( x - 2 )$

Choose the one alternative that best completes the statement or answers the question. A relation in x and y is given. Determine if the relation defines y as a one-to-one function of x. - $\{ ( - 6,8 ) , - 5 , - 8 ) , ( 4 , - 6 ) , ( 1,3 ) \}$

Solve the equation. Write the solution set with the exact values given in terms of natural or common logarithms. Also give approximate solutions to 4 decimal places, if necessary. - $5 ^ { 5 x + 6 } = 4 ^ { 8 x }$ ; use natural logarithms

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. - $2 \log _ { 6 } ( 6 p - 8 ) - 4 = - 2$

Use transformations of the graph $y = e ^ { x }$ to graph the function. Write the domain and range in interval notation. - $f ( x ) = - e ^ { x } + 1$

Choose the one alternative that best completes the statement or answers the question. Evaluate the function at the given value of x. Round to 4 decimal places if necessary. - $h ( x ) = \left( \frac { 1 } { 3 } \right) ^ { x } ; \quad h ( - 3 )$

Choose the one alternative that best completes the statement or answers the question. Solve for the indicated variable. - $A = P e ^ { r t }$ for $r$

Write the equation in logarithmic form. - $10 ^ { 5 } = 100,000$

Approximate the value of the logarithm to four decimal places. - $\log 512$

Write as the sum or difference of logarithms and fully simplify, if possible. Assume the variable represents a positive real number. - $\ln \sqrt [ 8 ] { \frac { e ^ { 7 } } { x ^ { 4 } + y } }$

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. - $\log \left( x ^ { 2 } - 2 x \right) = \log 15$

Determine whether the two functions are inverses. - $f ( x ) = 6 x + 6 \text { and } g ( x ) = \frac { 6 + x } { 6 }$

Equations and Inequalities

Functions and Relations

Polynomial and Rational Functions

Systems of Equations and Inequalities

Matrices and Determinants and Applications

Analytic Geometry

Sequences, Series, Induction, and Probability

Review of Prerequisites

Filters

Exam 4: Exponential and Logarithmic Functions

Solve the problem. -Use the graph of y=2−xy = 2 ^ { - x }y=2−x to graph the function . Write the domain and range in interval notation f(x)=2−xf ( x ) = 2 ^ { - x }f(x)=2−x

Solve the problem. -The enrollment at one Midwestern college is approximated by the formula P=3000(1.08)tP = 3000 ( 1.08 ) ^ { t }P=3000(1.08)t where ttt is the number of years after 2000 . What is the first year in which you would expect enrollment to surpass 3,700 ?

Simplify the expression without using a calculator. - log⁡5/4(256625)\log _ { 5 / 4 } \left( \frac { 256 } { 625 } \right)log5/4​(625256​)

Determine if the statement is true or false. - log⁡4(1p)=−log⁡4p\log _ { 4 } \left( \frac { 1 } { p } \right) = - \log _ { 4 } plog4​(p1​)=−log4​p

Solve the problem. -Use log⁡b3≈0.319\log _ { b } 3 \approx 0.319logb​3≈0.319 , use the properties of logarithms to approximate the following. Do not use a calcu log⁡b81\log _ { b } 81logb​81

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. - log⁡5(6x−14)=1+log⁡5(x−2)\log _ { 5 } ( 6 x - 14 ) = 1 + \log _ { 5 } ( x - 2 )log5​(6x−14)=1+log5​(x−2)

Solve the equation. Write the solution set with the exact values given in terms of natural or common logarithms. Also give approximate solutions to 4 decimal places, if necessary. - 55x+6=48x5 ^ { 5 x + 6 } = 4 ^ { 8 x }55x+6=48x ; use natural logarithms

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. - 2log⁡6(6p−8)−4=−22 \log _ { 6 } ( 6 p - 8 ) - 4 = - 22log6​(6p−8)−4=−2

Use transformations of the graph y=exy = e ^ { x }y=ex to graph the function. Write the domain and range in interval notation. - f(x)=−ex+1f ( x ) = - e ^ { x } + 1f(x)=−ex+1

Choose the one alternative that best completes the statement or answers the question. Evaluate the function at the given value of x. Round to 4 decimal places if necessary. - h(x)=(13)x;h(−3)h ( x ) = \left( \frac { 1 } { 3 } \right) ^ { x } ; \quad h ( - 3 )h(x)=(31​)x;h(−3)

Choose the one alternative that best completes the statement or answers the question. Solve for the indicated variable. - A=PertA = P e ^ { r t }A=Pert for rrr

Write the equation in logarithmic form. - 105=100,00010 ^ { 5 } = 100,000105=100,000

Approximate the value of the logarithm to four decimal places. - log⁡512\log 512log512

Write as the sum or difference of logarithms and fully simplify, if possible. Assume the variable represents a positive real number. - ln⁡e7x4+y8\ln \sqrt [ 8 ] { \frac { e ^ { 7 } } { x ^ { 4 } + y } }ln8x4+ye7​​

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. - log⁡(x2−2x)=log⁡15\log \left( x ^ { 2 } - 2 x \right) = \log 15log(x2−2x)=log15

Determine whether the two functions are inverses. - f(x)=6x+6 and g(x)=6+x6f ( x ) = 6 x + 6 \text { and } g ( x ) = \frac { 6 + x } { 6 }f(x)=6x+6 and g(x)=66+x​