Deck 5: Exponential and Logarithmic Functions

Question 1

Use the functions given by $f ( x ) = \frac { 1 } { 8 } x - 3$ and $g ( x ) = x ^ { 2 }$ to find the value $\left( f ^ { - 1 } \circ g ^ { - 1 } \right) ( 1 )$

A)32
B)35
C)33
D)37
E)30

Answer 1

32

Question 2

Show that $f$ and $g$ are functions by using the definition of inverse functions. $f ( x ) = \frac { 1 } { x } , g ( x ) = \frac { 1 } { x }$

A) $f ( g ( x ) ) = x , g ( f ( x ) ) = x - 1$
B) $f ( g ( x ) ) = x - 1 , g ( f ( x ) ) = x$
C) $f ( g ( x ) ) = x , g ( f ( x ) ) = x$
D) $f ( g ( x ) ) = x , g ( f ( x ) ) = x ^ { - 1 }$
E) $f ( g ( x ) ) = x ^ { - 1 } , g ( f ( x ) ) = x$

Answer 2

$f ( g ( x ) ) = x , g ( f ( x ) ) = x$

Question 3

Use the functions given by $f ( x ) = x + 4$ and $g ( x ) = 2 x - 6$ to find the composition of functions $g ^ { - 1 } \circ f ^ { - 1 }$

A) $\frac { x - 2 } { 2 }$
B) $\frac { x + 2 } { - 2 }$
C) $\frac { x - 2 } { - 2 }$
D) $\frac { - ( x + 2 ) } { 2 }$
E) $\frac { x + 2 } { 2 }$

Answer 3

$\frac { x + 2 } { 2 }$

Question 4

Determine whether the function has an inverse function, If it does, find its inverse function. $f ( x ) = | x - 2 | , x \leq 2$

A)No, $f$ does not have an inverse function.
B)Yes, $f$ has an inverse function, $f ^ { - 1 } ( x ) = 2 - x , x \geq 0$
C)Yes, $f$ has an inverse function, $f ^ { - 1 } ( x ) = - 2 - x , x \geq 0$
D)Yes, $f$ has an inverse function, $f ^ { - 1 } ( x ) = 2 + x , x \geq 0$
E)Yes, $f$ has an inverse function, $f ^ { - 1 } ( x ) = - 2 + x , x \geq 0$

Question 5

Use the functions given by $f ( x ) = x + 4$ and $g ( x ) = 2 x - 6$ to find the composition of functions $(g\circ f)^{-1}$

A) $\frac { - ( x + 2 ) } { - 2 }$
B) $\frac { - x + 2 } { 2 }$
C) $\frac { x - 2 } { 2 }$
D) $\frac { x - 2 } { - 2 }$
E) $\frac { x + 2 } { 2 }$

Question 6

Show that $f$ and $g$ are functions by using the definition of inverse functions. $f ( x ) = 5 x + 1 , g ( x ) = \frac { x - 1 } { 5 }$

A) $f ( g ( x ) ) = x , g ( f ( x ) ) = x$
B) $f ( g ( x ) ) = x ^ { - 1 } , g ( f ( x ) ) = x$
C) $f ( g ( x ) ) = x , g ( f ( x ) ) = x ^ { - 1 }$
D) $f ( g ( x ) ) = x + 1 , g ( f ( x ) ) = x$
E) $f ( g ( x ) ) = x , g ( f ( x ) ) = x + 1$

Question 7

Find the inverse function of the function f given by the set of ordered pairs. $\{ ( 6,2 ) , ( 5,3 ) , ( 4,4 ) , ( 3,5 ) \}$

A) $f ^ { - 1 } = \{ ( 2,6 ) , ( 3,5 ) , ( 4,4 ) , ( 5,3 ) \}$
B) $f ^ { - 1 } = \{ ( 2,6 ) , ( 3,5 ) , ( 4,4 ) , ( 5,5 ) \}$
C) $f ^ { - 1 } = \{ ( 2,6 ) , ( 3,5 ) , ( 4,2 ) , ( 5,3 ) \}$
D) $f ^ { - 1 } = \{ ( 2,6 ) , ( 3,3 ) , ( 4,4 ) , ( 5,3 ) \}$
E) $f ^ { - 1 } = \{ ( 2,2 ) , ( 3,5 ) , ( 4,4 ) , ( 5,3 ) \}$

Question 8

Sketch the graphs of inverse functions $f ( x ) = x ^ { 5 } - 2 , f ^ { - 1 } ( x ) = \sqrt [ 5 ] { x + 2 }$ in the same coordinate plane and show that the graphs are reflections of each other in the line $y = x$

A) $\begin{array} { l l r r } x & - 1 & 0 & 1 \\f ( x ) & - 3 & - 2 & - 1 \\\\x & - 4 & - 2 & - 1 \\f ^ { - 1 } ( x ) & - 1 & 0 & 1\end{array}$
B) $\begin{array} { l lrr } x & - 1 & 0 & 1 \\f ( x ) & - 3 & - 2 & - 1 \\\\x & - 3 & - 2 & -1\\f ^ { - 1 } ( x ) & - 1 & 2 & 1\end{array}$
C) $\begin{array} { l r rr } x & 0 & 0 & 1 \\f ( x ) & - 3 & - 2 & - 1 \\\\x & - 3 & - 2 & -1\\f ^ { - 1 } ( x ) & - 1 & 0 & 1\end{array}$
D) $\begin{array} { l l rr } x & - 1 & 0 & 1 \\f ( x ) & - 3 & - 2 & - 1 \\\\x & - 3 & - 2 & -1\\f ^ { - 1 } ( x ) & - 1 & 0 & 1\end{array}$
E)
$\begin{array}{llrr}x & -1 & 0 & -1 \\f(x) & -3 & -2 & -1 \\\\x & -3& 2& -1 \\f^{-1}(x) & -1 & -2 & 1\end{array}$

Question 9

Find the inverse function informally $f ( x ) = x - 5$ . Verify that $f \left( f ^ { - 1 } ( x ) \right) = x$ and $f ^ { - 1 } ( f ( x ) ) = x$

A) $f ^ { - 1 } ( x ) = x + 1$ , $f \left( f ^ { - 1 } ( x ) \right) = x$ , $f ^ { - 1 } ( f ( x ) ) = x$
B) $f ^ { - 1 } ( x ) = x + 5$ , $f \left( f ^ { - 1 } ( x ) \right) = x$ , $f ^ { - 1 } ( f ( x ) ) = x$
C) $f ^ { - 1 } ( x ) = x + 5$ , $f \left( f ^ { - 1 } ( x ) \right) = x$ , $f ^ { - 1 } ( f ( x ) ) = x - 1$
D) $f ^ { - 1 } ( x ) = x + 5$ , $f \left( f ^ { - 1 } ( x ) \right) = x + 1$ , $f ^ { - 1 } ( f ( x ) ) = x$
E) $f ^ { - 1 } ( x ) = x + 5$ , $f \left( f ^ { - 1 } ( x ) \right) = x$ , $f ^ { - 1 } ( f ( x ) ) = x ^ { - 1 }$

Question 10

Use the functions given by $f ( x ) = \frac { 1 } { 8 } x - 3$ and $g ( x ) = x ^ { 2 }$ to find the value $\left( g ^ { - 1 } \circ f ^ { - 1 } \right) ( - 3 )$

A)1
B)0
C)3
D)4
E)2

Question 11

Find the inverse function of the function f given by the set of ordered pairs. $\{ ( 1,4 ) , ( 2,5 ) , ( 3,6 ) , ( 4,7 ) \}$

A) $f ^ { - 1 } = \{ ( 4,4 ) , ( 5,2 ) , ( 6,3 ) , ( 7,4 ) \}$
B) $f ^ { - 1 } = \{ ( 4,1 ) , ( 5,5 ) , ( 6,3 ) , ( 7,4 ) \}$
C) $f ^ { - 1 } = \{ ( 4,1 ) , ( 5,2 ) , ( 6,6 ) , ( 7,4 ) \}$
D) $f ^ { - 1 } = \{ ( 4,1 ) , ( 5,2 ) , ( 6,3 ) , ( 7,4 ) \}$
E) $f ^ { - 1 } = \{ ( 4,1 ) , ( 5,2 ) , ( 6,3 ) , ( 7,7 ) \}$

Question 12

Find the inverse function informally $f ( x ) = 2 x$ . Verify that $f \left( f ^ { - 1 } ( x ) \right) = x$ and $f ^ { - 1 } ( f ( x ) ) = x$

A) $f ^ { - 1 } ( x ) = \frac { x } { 2 }$ , $f \left( f ^ { - 1 } ( x ) \right) = x ^ { - 1 }$ , $f ^ { - 1 } ( f ( x ) ) = x$
B) $f ^ { - 1 } ( x ) = \frac { x } { 2 }$ , $f \left( f ^ { - 1 } ( x ) \right) = 2 x$ , $f ^ { - 1 } ( f ( x ) ) = 3 x$
C) $f ^ { - 1 } ( x ) = \frac { x } { 2 }$ , $f \left( f ^ { - 1 } ( x ) \right) = x$ , $f ^ { - 1 } ( f ( x ) ) = 2 x$
D) $f ^ { - 1 } ( x ) = \frac { x } { 2 }$ , $f \left( f ^ { - 1 } ( x ) \right) = 3 x$ , $f ^ { - 1 } ( f ( x ) ) = x$
E) $f ^ { - 1 } ( x ) = \frac { x } { 2 }$ , $f \left( f ^ { - 1 } ( x ) \right) = x$ , $f ^ { - 1 } ( f ( x ) ) = x$

Question 13

Use the graph of $f$ to complete the table and to sketch the graph of $f ^ { - 1 }$ $\begin{array}{llllll}x & 0 & 1 & 2 & 3 & 4\\f^{-1}(x)\end{array}$

A) $\begin{array} { c c c c c c } x & 0 & 1 & 2 & 3 & 4 \\f ^ { - 1 } ( x ) & 0 & 4 & - 2 & 2 & 1\end{array}$
B) $\begin{array} { c c c c c c } x & 0 & 1 & 2 & 3 & 4 \\f ^ { - 1 } ( x ) & 4 & - 2 & 2 & 0 & 1\end{array}$
C) $\begin{array} { l l l l l l } x & 0 & 1 & 2 & 3 & 4 \\f ^ { - 1 } ( x ) & - 2 & 0 & 1 & 2 & 4\end{array}$
D) $\begin{array} { c c c c c c } x & 0 & 1 & 2 & 3 & 4 \\f ^ { - 1 } ( x ) & 2 & 4 & - 2 & 0 & 1\end{array}$
E) $\begin{array} { c c c c c c } x & 0 & 1 & 2 & 3 & 4 \\f ^ { - 1 } ( x ) & 1 & 2 & - 2 & 4 & 0\end{array}$

Question 14

Determine whether the function has an inverse function, If it does, find its inverse function. $f ( x ) = 36 + x ^ { 2 } , x \leq 0$

A)Yes, $f$ has an inverse function, $f ^ { - 1 } ( x ) = - \sqrt { x - 36 } , x \geq 36$
B)No, $f$ does not have an inverse function.
C)Yes, $f$ has an inverse function, $f ^ { - 1 } ( x ) = - \sqrt { x + 36 } , x \geq 36$
D)Yes, $f$ has an inverse function, $f ^ { - 1 } ( x ) = \sqrt { x - 36 } , x \geq 36$
E)Yes, $f$ has an inverse function, $f ^ { - 1 } ( x ) = - \sqrt { x - 36 } , x \leq 36$

Question 15

Use the graph of $f$ to determine whether the function has an inverse function.

A)Yes, $f$ has an inverse function.
B)No, $f$ does not have an inverse function.

Question 16

A company's profit $P$ for producing $x$ units is given by $P ( x ) = 47 x - 5736$ . Find the inverse function $P ^ { - 1 } ( x )$ and explain what it represents. Describe the domains of $P ( x )$ and $P ^ { - 1 } ( x )$ .

A) $P ^ { - 1 } ( x ) = \frac { x + 5736 } { 47 }$ , $P ^ { - 1 } ( x )$ represents the number of units that must be sold to obtain the profit of $x$ .Domain of $P ( x ) : [ 0,10 )$ , Domain of $P ^ { - 1 } ( x ) : [ - 5736 , \infty )$
B) $P ^ { - 1 } ( x ) = \frac { x + 5736 } { 47 }$ , $P ^ { - 1 } ( x )$ represents the number of units that must be sold to obtain the profit of $x$ .Domain of $P ( x ) : [ 0 , \infty )$ , Domain of $P ^ { - 1 } ( x ) : [ - 5736,5736 )$
C) $P ^ { - 1 } ( x ) = \frac { x + 5736 } { 47 }$ , $P ^ { - 1 } ( x )$ represents the number of units that must be sold to obtain the profit of $x$ .Domain of $P ( x ) : [ 0 , \infty )$ , Domain of $P ^ { - 1 } ( x ) : [ - 5736 , \infty )$
D) $P ^ { - 1 } ( x ) = \frac { x + 5736 } { 47 }$ , $P ^ { - 1 } ( x )$ represents the number of units that must be sold to obtain the profit of $x$ .Domain of $P ( x ) : [ 1 , \infty )$ , Domain of $P ^ { - 1 } ( x ) : [ - 5736 , \infty )$
E) $P ^ { - 1 } ( x ) = \frac { x + 5736 } { 47 }$ , $P ^ { - 1 } ( x )$ represents the number of units that must be sold to obtain the profit of $x$ .Domain of $P ( x ) : [ 0 , \infty )$ , Domain of $P ^ { - 1 } ( x ) : [ - 5736,47 )$

Question 17

Sketch the graphs of inverse functions $f ( x ) = \frac { x } { 3 } , f ^ { - 1 } ( x ) = 3 x$ in the same coordinate plane and show that the graphs are reflections of each other in the line $y = x$

A) $\begin{array} { l l l l } x & - 3 & 0 & 3 \\f ( x ) & - 1 & 0 & 1\end{array}$ $\begin{array} { l l l l } x & - 1 & 0 & 1 \\f ^ { - 1 } ( x ) & - 3 & 0 & 3\end{array}$
B) $\begin{array} { l l l l } x & - 2 & 0 & 3 \\f ( x ) & - 1 & 0 & 1\end{array}$ $\begin{array} { l l l l } x & - 1 & 0 & 1 \\f ^ { - 1 } ( x ) & - 3 & 0 & 3\end{array}$
C) $\begin{array} { l l l l } x & - 3 & 0 & 3 \\f ( x ) & - 1 & 0 & 1\end{array}$ $\begin{array} { l l l l } x & - 1 & 0 & 1 \\f ^ { - 1 } ( x ) & - 3 & 2 & 3\end{array}$
D) $\begin{array} { l l l l } x & - 3 & 0 & 0 \\f ( x ) & - 1 & 0 & 1\end{array}$ $\begin{array} { l l l l } x & - 1 & 0 & 1 \\f ^ { - 1 } ( x ) & - 3 & 0 & 3\end{array}$
E) $\begin{array} { l l l l } x & - 3 & 0 & 3 \\f ( x ) & - 1 & 0 & 1\end{array}$ $\begin{array} { l l l l } x & - 1 & 0 & 1 \\f ^ { - 1 } ( x ) & - 2 & 0 & 3\end{array}$

Question 18

Determine whether the function has an inverse function, If it does, find its inverse function. $f ( x ) = x ^ { 4 }$

A)No, $f$ does not have an inverse function.
B)Yes, $f$ has an inverse function, $y = x ^ { 4 }$
C)Yes, $f$ has an inverse function, $y = - x ^ { - 4 }$
D)Yes, $f$ has an inverse function, $y = x ^ { - 4 }$
E)Yes, $f$ has an inverse function, $y = - x ^ { 4 }$

Question 19

Use the graph of $f$ to determine whether the function has an inverse function.

A)No, $f$ does not have an inverse function.
B)Yes, $f$ has an inverse function.

Question 20

Use the functions given by $f ( x ) = \frac { 1 } { 8 } x - 3$ and $g ( x ) = x ^ { 2 }$ to find the value $\left( g ^ { - 1 } \circ g ^ { - 1 } \right) ( - 4 )$

A)16
B)-16
C)32
D)The value does not exist.
E)-32

Question 21

Use a calculator to evaluate $( 2.6 )$ . Round your result to three decimal places.

A) $\approx 0.463$
B) $\approx 4.463$
C) $\approx 3.463$
D) $\approx 1.463$
E) $\approx 2.463$

Question 22

Sketch the graph of the function $y = e ^ { - 0.1 x }$

A)
B)
C)
D)
E)

Question 23

Use a calculator to evaluate the function $f ( x ) = e ^ { x }$ for the given value of $x$ , $x = 0.3$ . Round your result to three decimal places.

A) $e ^ { 0.3 } \approx 1.050$
B) $e ^ { 0.3 } \approx 0.050$
C) $e ^ { 0.3 } \approx 1.50$
D) $e ^ { 0.3 } \approx 0.350$
E) $e ^ { 0.3 } \approx 1.350$

Question 24

You deposit a lump sum $P$ in a trust fund on the day your child is born. The fund earns 6.5% interest compounded continuously. Find the amount $P$ that will yield the given balance $A$ on your child's 25th birthday. $A = \$ 100,000$

A) $P \approx \$ 10,691.17$
B) $P \approx \$ 9,691.17$
C) $P \approx \$ 14,691.17$
D) $P \approx \$ 11,691.17$
E) $P \approx \$ 19,691.17$

Question 25

Sketch the graph of the function $f ( x ) = \left( \frac { 1 } { 5 } \right) ^ { x }$ .

A)
B)
C)
D)
E)

Question 26

Evaluate the expression below. Round your results to three decimal places. $e ^ { 3 }$

A)2.718
B)20.086
C)0.135
D)54.598
E)7.389

Question 27

Match the function $f ( x ) = 2 ^ { x - 3 }$ with its graph.

A)
B)
C)
D)
E)

Question 28

Strontinum-90 has a half life of 29.1 years.The amount S of 100 kilograms of Strontinum6990 present after t years is given by $S = 100 e ^ { - 0.0238 t }$ How much of the 100 kilograms will remain after 50 years?

A)about 31.4 kilograms
B)about 35.4 kilograms
C)about 30.4 kilograms
D)about 37.4 kilograms
E)about 40.4 kilograms

Question 29

The present value of money is the principal $P$ you need to invest today so that it will grow to an amount $A$ at the end of specified time. The present value formula $P = A \left( 1 + \frac { r } { n } \right) ^ { - n t }$ is obtained by solving the compound interest formula $A = P \left( 1 + \frac { r } { n } \right) ^ { n t }$ for $P$ . Recall that $t$ is the number of years, $r$ is the interest rate per year, and $n$ is the number of compoundings per year. find the present value of amount $A$ invested at rate $r$ for $t$ years, compounded $n$ times per year. $A = \$ 10,000 , r = 6 \% , t = 5 \text { years } , n = 4$

A) $P \approx 6424.70$
B) $P \approx 3424.70$
C) $P \approx 4424.70$
D) $P \approx 7424.70$
E) $P \approx 5424.70$

Question 30

The present value of money is the principal $P$ you need to invest today so that it will grow to an amount $A$ at the end of specified time. The present value formula $P = A \left( 1 + \frac { r } { n } \right) ^ { - n t }$ is obtained by solving the compound interest formula $A = P \left( 1 + \frac { r } { n } \right) ^ { n t }$ for $P$ . Recall that $t$ is the number of years, $r$ is the interest rate per year, and $n$ is the number of compoundings per year. find the present value of amount $A$ invested at rate $r$ for $t$ years, compounded $n$ times per year. $A = \$ 1,000,000 , r = 8 \% , t = 20 \text { years }, n = 2$

A) $P \approx \$ 108,289.04$
B) $P \approx \$ 208,289.04$
C) $P \approx \$ 308,289.04$
D) $P \approx \$ 20,289.04$
E) $P \approx \$ 1008,289.04$

Question 31

Use a calculator to evaluate $6 ^ { - \sqrt { 2 } }$ . Round your result to three decimal places.

A) $\approx 0.079$
B) $\approx 4.079$
C) $\approx 1.079$
D) $\approx 3.079$
E) $\approx 2.079$

Question 32

The demand function for a limited edition comic book is given by $p = 3000 \left( 1 - \frac { 5 } { 5 + e ^ { - 0.015 x } } \right)$ Find the price $p$ for a demand of $x = 45$ units.

A)$5722.74
B)$277.26
C)$369.89
D)$340.13
E)$6369.89

Question 33

Sketch the graph of the function $y = 3 ^ { - x ^ { 2 } }$ .

A)
B)
C)
D)
E)

Question 34

Sketch the graph of the function $g ( x ) = 4 ^ { x }$ .

A)
B)
C)
D)
E)

Question 35

Sketch the graph of the function $N ( t ) = 2 - e ^ { t }$

A)
B)
C)
D)
E)

Question 36

The number of certain type of bacteria increases according to the model $P ( t ) = 100 e ^ { 0.01896 t }$ where t is time (in hours)
a)Find P(0).
b)Find P(5).
c)Find P(10).
d)Find P(24).

A)a) $P ( 0 ) = 100$ b) $P ( 5 ) \approx 109.94$ c) $P ( 10 ) \approx 120.88$ d) $P ( 24 ) \approx 157.62$
B)a) $P ( 0 ) \approx 100$ b) $P ( 5 ) \& 119.94$ c) $P ( 10 ) \approx 102.88$ d) $\mathrm { P } ( 24 ) \approx 257.62$
C)a) $\mathrm { P } ( 0 ) = 120.56$ b) $P ( 5 ) = 100.94$ c) $P ( 10 ) \approx 620.88$ d) $P ( 24 ) \approx 517.62$
D)a) $\mathrm { P } ( 0 ) = 109.78$ b) $P ( 5 ) \approx 100.94$ c) $P ( 10 ) \approx 150.88$ d) $P ( 24 ) \approx 357.62$
E)a) $P ( 0 ) = 105$ b) $P ( 5 ) \approx 119.94$ c) $P ( 10 ) \approx 120.08$ d) $P ( 24 ) \approx 157.52$

Question 37

Determine whether $e = \frac { 271,801 } { 99,990 }$ . Justify your answer.

A)Yes, $e = \frac { 271,801 } { 99,990 }$ because $e$ is a rational number.
B)No, $e \neq \frac { 271,801 } { 99,990 }$ because $e$ is not a rational number.

Question 38

Match the function $f ( x ) = 2 ^ { - x }$ with its graph.

A)
B)
C)
D)
E)

Question 39

Determine which of the following functions is graphed below. $2 ^ { x } - 1,2 ^ { - x } - 1,2 ^ { - x } + 1,2 ^ { - x } + 2,2 ^ { x } + 2$

A) $2 ^ { x } - 1$
B) $2 ^ { - x } - 1$
C) $2 ^ { - x } + 1$
D) $2 ^ { - x } + 2$
E) $2 ^ { x } + 2$

Question 40

Use a calculator to evaluate the function $f ( x ) = e ^ { x }$ for the given value of $x$ , $x = 4$ . Round your result to three decimal places.

A) $e ^ { 4 } \approx 54.598$
B) $e ^ { 4 } \approx 44.598$
C) $e ^ { 4 } \approx 50.598$
D) $e ^ { 4 } \approx 52.598$
E) $e ^ { 4 } \approx 45.598$

Question 41

Find the domain of the function $k ( x ) = \log _ { 8 } ( 4 - x )$ .

A) $( 4 , \infty )$
B) $( - \infty , 8 )$
C) $( - 4 , \infty )$
D) $( - \infty , 4 )$
E) $( 8 , \infty )$

Question 42

Rewrite the exponential equation $3 ^ { - 2 } = \frac { 1 } { 9 }$ in logarithmic form.

A) $\log _ { 3 } \frac { 1 } { 9 } = - 2$
B) $\log _ { 2 } 9 = - 2$
C) $\log _ { 3 } 9 = - 2$
D) $\log _ { 9 } 3 = - 2$
E) $\log _ { 3 } \frac { 1 } { 9 } = 2$

Question 43

Rewrite the logarithmic equation $\log _ { 6 } \frac { 1 } { 36 } = - 2$ in exponential form.

A) $6 ^ { 36 } = - 2$
B) $6 ^ { 1 / 36 } = - 2$
C) $6 ^ { - 2 } = \frac { 1 } { 36 }$
D) $\left( \frac { 1 } { 36 } \right) ^ { - 2 } = 6$
E) $6 ^ { - 2 } = - \frac { 1 } { 36 }$

Question 44

Expand the given logarithmic expression. Assume all variable expressions represent positive real numbers. $\ln \left( x ^ { 7 / 4 } y ^ { 3 / 2 } \right)$

A) $\frac { 21 } { 8 } \ln x y$
B) $\frac { 21 } { 8 } \ln x \ln y$
C) $x \ln \left( \frac { 7 } { 4 } \right) + y \ln \left( \frac { 3 } { 2 } \right)$
D) $\frac { 3 } { 2 } \ln x + \frac { 7 } { 4 } \ln y$
E) $\frac { 7 } { 4 } \ln x + \frac { 3 } { 2 } \ln y$

Question 45

Evaluate the logarithm $\log _ { 7 } 714$ using the change of base formula. Round to 3 decimal places.

A)6.571
B)0.296
C)3.377
D)12.786
E)2.854

Question 46

Expand the logarithmic expression $\ln \left( \frac { z ^ { 6 } } { \sqrt [ 5 ] { x ^ { 4 } y ^ { 4 } } } \right)$ . Assume all variable expressions represent positive real numbers.

A) $\ln ( 6 z - 20 x + 20 y )$
B) $6 \ln z - \frac { 4 } { 5 } \ln x - \frac { 4 } { 5 } \ln y$
C) $6 \ln z - 20 \ln x - 20 \ln y$
D) $\ln \left( 6 z - \frac { 4 } { 5 } x - \frac { 4 } { 5 } y \right)$
E) $6 \ln z - \frac { 4 } { 5 } \ln x + \frac { 4 } { 5 } \ln y$

Question 47

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

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

Question 48

Write the exponential equation $e ^ { 3 / 2 } = 4.4817 \ldots$ in logarithmic form.

A) $2.303 \log \left( \frac { 3 } { 2 } \right) = 4.4817 \ldots$
B) $\log _ { 10 } ( 4.4817 \ldots ) = \frac { 3 } { 2 }$
C) $\ln \left( \frac { 3 } { 2 } \right) = 4.4817 \ldots$
D) $\ln ( 4.4817 \ldots ) = \frac { 3 } { 2 }$
E) $\ln 3 = \frac { 4.4817 \ldots } { 2 }$

Question 49

Write the expression below as a single logarithm with a coefficient of 1. Assume all variable expressions represent positive real numbers. $5 \log _ { 2 } t - \frac { 1 } { 6 } \log _ { 2 } u + 4 \log _ { 2 } v$

A) $\log _ { 2 } \left( t ^ { 5 } - u ^ { - 6 } + v ^ { 4 } \right)$
B) $\log _ { 2 } \left( t ^ { 5 } + \frac { 1 } { \sqrt [ 6 ] { u } } + v ^ { 4 } \right)$
C) $\log _ { 2 } \left( \frac { t ^ { 5 } v ^ { 4 } } { \sqrt [ 6 ] { u } } \right)$
D) $\log _ { 2 } \left( \frac { t ^ { 5 } v ^ { 6 } } { \sqrt [ 4 ] { u } } \right)$
E) $\log _ { 2 } \left( \frac { t ^ { 6 } v ^ { 4 } } { \sqrt [ 5 ] { u } } \right)$

Question 50

Approximate the logarithm below using the properties of logarithms, given $\log _ { b } 2 \approx 0.3562,$ $\log _ { b } 3 \approx 0.5646,$ and $\log _ { b } 5 \approx 0.8271.$ $\log _ { b } \frac { 3 } { 2 }$

A)-0.2084
B)-0.2625
C)0.2084
D)0.2625
E)0.4709

Question 51

Condense the expression $\frac { 1 } { 5 } \left[ \log _ { 5 } x + \log _ { 5 } 6 \right] - \left[ \log _ { 5 } y \right]$ to the logarithm of a single term.

A) $\log _ { 5 } \frac { ( 6 x ) ^ { 5 } } { y }$
B) $\log _ { 5 } \frac { 6 x } { 5 y }$
C) $\log _ { 5 } \sqrt [ 5 ] { \frac { 6 x } { y } }$
D) $\log _ { 5 } \frac { \sqrt [ 5 ] { 6 x } } { y }$
E) $\log _ { 5 } \sqrt[5] { 6 x } - \log _ { 5 } y$

Question 52

Write the exponential equation $3 ^ { 2 } = 9$ in its logarithmic form.

A) $\log _ { 2 } 9 = 3$
B) $\log _ { 3 } 2 = 9$
C) $\log _ { 9 } 3 = 2$
D) $\log _ { 2 } 3 = 9$
E) $\log _ { 3 } 9 = 2$

Question 53

Write the logarithmic equation $4 = \log _ { 2 } 16$ in its exponential form.

A) $2 ^ { 16 } = 4$
B) $2 ^ { 4 } = 16$
C) $16 ^ { 4 } = 2$
D) $4 ^ { 2 } = 16$
E) $4 ^ { 16 } = 2$

Question 54

Condense the expression $\log _ { 3 } x + \log _ { 3 } 4$ to the logarithm of a single term.

A) $\log ( 4 x ) ^ { 3 }$
B) $\log _ { 3 } 4 x$
C) $\log _ { 3 } 4 ^ { x }$
D) $\log _ { 3 } x ^ { 4 }$
E) $\log _ { 3 } ( x + 4 )$

Question 55

Match the function below with its graph. $f ( x ) = 3 \ln x - 2$ Graph I : Graph IV: Graph II: Graph V: Graph III:

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

Question 56

Write $5 \ln ( x + 7 ) - 3 \ln x - 4 \ln \left( x ^ { 2 } + 8 \right)$ as a single logarithm with a coefficient of 1. Assume all variable expressions represent positive real numbers.

A) $\ln \left( x ^ { 3 } ( x + 7 ) ^ { 5 } \left( x ^ { 2 } + 8 \right) ^ { 4 } \right)$
B) $\ln \left( \frac { 5 ( x + 7 ) } { 12 x \left( x ^ { 2 } + 8 \right) } \right)$
C) $\ln \left( 60 x ( x + 7 ) \left( x ^ { 2 } + 8 \right) \right)$
D) $\ln \left( - 4 x ^ { 2 } + 2 x + 3 \right)$
E) $\ln \left( \frac { ( x + 7 ) ^ { 5 } } { x ^ { 3 } \left( x ^ { 2 } + 8 \right) ^ { 4 } } \right)$

Question 57

Find the vertical asymptote of the logarithmic function below. $f ( x ) = \log _ { 2 } ( x - 4 )$

A) $x = 5$
B) $x = 2$
C) $x = - 5$
D) $x = - 4$
E) $x = 4$

Question 58

Use the properties of logarithms to simplify the logarithmic expression below. $\log _ { 5 } \sqrt { 175 }$

A) $2 \log _ { 5 } 7$
B) $1 + \frac { 1 } { 2 } \log _ { 5 } 7$
C) $2 + \frac { 1 } { 2 } \log _ { 5 } 7$
D) $2 + \log _ { 5 } 7$
E) $1 + \log _ { 5 } 7$

Question 59

Evaluate the logarithm $\log _ { 2 } \frac { 1 } { 32 }$ without using a calculator.

A) $- 5$
B) $4$
C) $16$
D) $- 4$
E) $5$

Question 60

Write the logarithmic equation $\ln 6 = 1.792 \ldots$ in exponential form.

A) $e ^ { 1.792 \ldots } = 6$
B) $10 ^ { 6 } = 1.792 \ldots$
C) $2.303 e ^ { 1.792 \ldots } = 6$
D) $2.303 \times 10 ^ { 6 } = 1.792 \ldots$
E) $e ^ { 6 } = 1.792 \ldots$

Question 61

Approximate the solution to $\ln 5 x = 3.2$ . Round to 3 decimal places.

A)1.896
B)4.907
C)4.809
D)-0.446
E)316.979

Question 62

Solve for $x.$ $\ln ( 6 x - 11 ) = 0$

A) $x = 4$
B) $x = 2$
C) $x = 5$
D) $x = \frac { 11 } { 6 }$
E) $x = \frac { 7 } { 2 }$

Question 63

Solve the exponential equation algebraically. Approximate the result to three decimal places. $e ^ { 2 x } - 5 e ^ { x } + 4 = 0$

A) $x = 0.000,1.000$
B) $x = 0.000,1.386$
C) $x = 0.000,4.000$
D) $x = 1.000,4.000$
E) $x = 1.386,4.000$

Question 64

Use algebraic procedures to find the exact solution of the equation $\ln x = \frac { 1 } { 2 } \ln \left( 2 x + \frac { 19 } { 9 } \right) + \frac { 1 } { 2 } \ln 9$ .

A)-1, 19
B) $- \frac { 19 } { 14 }$
C)2, 9
D)19
E)1

Question 65

Approximate the solution to $\ln ( x + 3 ) - \ln x = 1$ . Round to 3 decimal places.

A)-0.632
B)0.250
C)1.746
D)2.718
E)0.333

Question 66

Solve the exponential equation algebraically. Approximate the result to three decimal places. $\frac { 400 } { 6 + e ^ { 6 x } } = 8$

A) $x = 0.671$
B) $x = \inf$
C) $x = 0.631$
D) $x = 1.001$
E) $x = 0.996$

Question 67

Use algebraic procedures to find the exact solution of the equation $\log _ { 5 } x + \log _ { 5 } ( x - 20 ) = 3$ .

A)0, -20
B)25
C)3, 5
D)-5, 25
E)5

Question 68

Use algebraic procedures to find the exact solution(s) of the equation below. $6 ^ { 9 x + 2 } = \frac { 1 } { 216 }$

A) $x = - \frac { 5 } { 9 }$
B) $x = \frac { 1 } { 9 }$
C) $x = \frac { 1 } { 6 }$
D) $x = - \frac { 5 } { 6 }$
E) $x = \frac { 5 } { 6 }$

Question 69

Solve $\left( \frac { 1 } { 2 } \right) ^ { x } = 8$ for x.

A)1
B) $- 1$
C) $- 3$
D) $- 2$
E)no solution

Question 70

The approximate lengths and diameters (in inches) of common nails are shown in the table. Find a logarithmic equation that relates the diameter $y$ of a common nail to its length $$x.$$ $$\begin{array} { | c | c | } \hline \text { Length, } x & \text { Diameter, } y \\\hline 1 & 0.095 \\\hline 2 & 0.156 \\\hline 3 & 0.208 \\\hline\end{array}$$ $$\begin{array} { | c | c | } \hline \text { Length, } x & \text { Diameter, } y \\\hline 4 & 0.256 \\\hline 5 & 0.300 \\\hline 6 & 0.342 \\\hline\end{array}$$

A) $\ln y = - 0.095 \ln x + \ln 0.715$
B) $\ln y = 0.095 \ln x - \ln 0.715$
C) $\ln y = 0.715 \ln x - \ln 0.095$
D) $\ln y = - 0.095 \ln x - \ln 0.715$
E) $\ln y = 0.715 \ln x + \ln 0.095$

Question 71

Use algebraic procedures to find the exact solution(s) of the equation below. $10 ^ { 3 - x } = 370$

A) $x = 1 + 3 \log 370$
B) $x = - 1 - 3 \log 370$
C) $x = - 1 + 3 \log 370$
D) $x = - 3 - \log 370$
E) $x = 3 - \log 370$

Question 72

An initial investment of $4000 grows at an annual interest rate of 4% compounded continuously. How long will it take to double the investment?

A)17.33 years
B)18.33 years
C)18.00 years
D)17.00 years
E)1 year

Question 73

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

A)9.803
B)15.777
C)20.606
D)-20.606
E)-3.268

Question 74

Solve for x: $9 \left( 10 ^ { x - 3 } \right) = 23$ . Round to 3 decimal places.

A)3.407
B)0.407
C)1.362
D)-1.362
E)no solution

Question 75

The average monthly sales $y$ (in billions of dollars) in retail trade in the United States from 1996 to 2005 can be approximated by the model $y = 22 + 117 \ln t,$ $6 \leq t \leq 15$ where $t$ represents the year, with $t = 6$ corresponding to 1996. Estimate the year in which the average monthly sales first exceeded $310 billion.

A)2001
B)2004
C)2002
D)2000
E)1996

Question 76

An industrial psychologist has determined that the average percent score for an employee on a test of the employee's knowledge of the company's product is given by $P = \frac { 100 } { 1 + 24 e ^ { - 0.15 t } }$ where t is the number of weeks on the job and P is the percent score. Estimate (to the nearest week) the expected number of weeks of employment that are necessary for an employee to earn a 85% score on the test.

A)38 weeks
B)37 weeks
C)42 weeks
D)60 weeks
E)33 weeks

Question 77

Apply the Inverse Property of logarithmic or exponential functions to simplify the expression below. $\log _ { 8 } 64 ^ { 2 x + 5 }$

A) $4 x + 10$
B) $16 x + 40$
C) $8 ^ { 2 x + 5 }$
D) $2 \log _ { 8 } ( 2 x + 5 )$
E) $8 \log _ { 8 } ( 2 x + 5 )$

Question 78

Approximate the solution of $16 e ^ { 7 x } = 22$ to 3 decimal places. (You may use a graphing utility.)

A)-6.682
B)0.164
C)-1.627
D)2.229
E)0.045

Question 79

Use a graphing utility to approximate the solution to $\log _ { 4 } x + \log _ { 4 } ( 2 x + 1 ) = 2$ . Round to 3 decimal places.

A)6.179
B)5.179
C)3.089
D)2.589
E)no solution

Question 80

Solve for x. $2 ^ { 3 x } = 128$

A) $\frac { 128 } { 3 }$
B) $- \frac { 64 } { 3 }$
C) $\frac { 7 } { 3 }$
D) $\frac { 3 } { 7 }$
E)2