Question 1

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

Accepted Answer

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

Question 2

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.

Accepted Answer

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)  $e ^ { 0.3 } \approx 1.350$ 
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 3

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

Accepted Answer

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)  $f ^ { - 1 } ( x ) = \frac { x } { 2 }$ , $f \left( f ^ { - 1 } ( x ) \right) = x$ , $f ^ { - 1 } ( f ( x ) ) = x$ 
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 4

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

Accepted Answer

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

Question 5

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

Accepted Answer

A)  $\log ( 4 x ) ^ { 3 }$ 
B)  $\log _ { 3 } 4 x$ 
C)  $\log _ { 3 } 4 ^ { x }$ 
D)  $\log _ { 3 } x ^ { 4 }$ 
E)  $\log _ { 3 } ( x + 4 )$ 
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 6

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

Accepted Answer

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

Question 7

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

Accepted Answer

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

Question 8

A stock analyst attempts to express the price p of a share of XYZ stock as an exponentially increasing function of the time since XYZ's initial public offering (IPO): $p = 10 e ^ { k m }$ , where m is the number of months since the IPO. The price of a share was $10.00 at the time of the IPO and $12.80 four months after the IPO. What is the approximate value of k? Round to the nearest thousandth.

Accepted Answer

A) 0.637 per month 
B) 0.062 per month 
C) 3.200 per month 
D) 1.163 per month 
E) 16.204 per month 
A) 0.637 per month 
B) 0.062 per month 
C) 3.200 per month 
D) 1.163 per month 
E) 16.204 per month

Question 9

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

Accepted Answer

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

Question 10

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.

Accepted Answer

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

Question 11

The decibel (dB) is defined as $\mathrm { dB } = 10 \log \frac { P _ { 2 } } { P _ { 1 } }$ , where P2 is the power of a particular signal and P1 is the power of some reference signal. In the case of sounds, the reference signal is a sound level that is just barely audible. How many dBs does a sound have if its power is 7,320,000 times that of the reference sound? Round to the nearest tenth.

Accepted Answer

A) 158.1 dB 
B) 68.6 dB 
C) 29.8 dB 
D) 10.0 dB 
E) 0.0 dB 
A) 158.1 dB 
B) 68.6 dB 
C) 29.8 dB 
D) 10.0 dB 
E) 0.0 dB

Question 12

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

Accepted Answer

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

Question 13

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

Accepted Answer

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

Question 14

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).

Accepted Answer

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) $P ( 24 ) \approx 157.52$ 
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 15

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

Accepted Answer

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

Question 16

An investment is expected to pay 7% per year compounded continuously. If you want the value of the investment to be $600,000 after 25 years, how much should you invest initially? Round to the nearest dollar.

Accepted Answer

A) $26,066 
B) $52,132 
C) $104,264 
D) $208,529 
E) $267,494 
A) $26,066 
B) $52,132 
C) $104,264 
D) $208,529 
E) $267,494

Question 17

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

Accepted Answer

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, $f ^ { - 1 } ( x ) = - \sqrt { x - 36 } , x \leq 36$ 
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 18

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

Accepted Answer

A)  $\log _ { 3 } \frac { 1 } { 9 } = - 2$ 
B)  $\log _ { 2 } 9 = - 2$ 
C)  $\log _ { 3 } 9 = - 2$ 
D)  $\log _ { 9 } 3 = - 2$ 
E)  $\log _ { 3 } \frac { 1 } { 9 } = 2$ 
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 19

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

Accepted Answer

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)  $\log _ { 2 } \left( \frac { t ^ { 6 } v ^ { 4 } } { \sqrt [ 5 ] { u } } \right)$ 
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 20

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

Accepted Answer

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

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

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.

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

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

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

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

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

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

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

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

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).

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

An investment is expected to pay 7% per year compounded continuously. If you want the value of the investment to be $600,000 after 25 years, how much should you invest initially? Round to the nearest dollar.

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

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

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

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Exam 5: Exponential and Logarithmic Functions

Use algebraic procedures to find the exact solution of the equation log⁡5x+log⁡5(x−20)=3\log _ { 5 } x + \log _ { 5 } ( x - 20 ) = 3log5​x+log5​(x−20)=3 .

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

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

Evaluate the expression below. Round your results to three decimal places. e3e ^ { 3 }e3

Condense the expression log⁡3x+log⁡34\log _ { 3 } x + \log _ { 3 } 4log3​x+log3​4 to the logarithm of a single term.

Solve the exponential equation algebraically. Approximate the result to three decimal places. 4006+e6x=8\frac { 400 } { 6 + e ^ { 6 x } } = 86+e6x400​=8

Sketch the graph of the function f(x)=(15)xf ( x ) = \left( \frac { 1 } { 5 } \right) ^ { x }f(x)=(51​)x .

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

Use a calculator to evaluate 6−26 ^ { - \sqrt { 2 } }6−2​ . Round your result to three decimal places.

Approximate the solution to ln⁡5x=3.2\ln 5 x = 3.2ln5x=3.2 . Round to 3 decimal places.

The number of certain type of bacteria increases according to the model P(t)=100e0.01896tP ( t ) = 100 e ^ { 0.01896 t }P(t)=100e0.01896t where t is time (in hours) a)Find P(0). b)Find P(5). c)Find P(10). d)Find P(24).

Evaluate the logarithm log⁡7714\log _ { 7 } 714log7​714 using the change of base formula. Round to 3 decimal places.

An investment is expected to pay 7% per year compounded continuously. If you want the value of the investment to be $600,000 after 25 years, how much should you invest initially? Round to the nearest dollar.

Determine whether the function has an inverse function, If it does, find its inverse function. f(x)=36+x2,x≤0f ( x ) = 36 + x ^ { 2 } , x \leq 0f(x)=36+x2,x≤0

Rewrite the exponential equation 3−2=193 ^ { - 2 } = \frac { 1 } { 9 }3−2=91​ in logarithmic form.

Write the expression below as a single logarithm with a coefficient of 1. Assume all variable expressions represent positive real numbers. 5log⁡2t−16log⁡2u+4log⁡2v5 \log _ { 2 } t - \frac { 1 } { 6 } \log _ { 2 } u + 4 \log _ { 2 } v5log2​t−61​log2​u+4log2​v

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

Fundamental Concepts of Algebra

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Limits and Derivatives

Applications of the Derivative

Further Applications of the Derivative

Derivatives of Exponential and Logarithmic Functions

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

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.

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

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

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

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

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

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

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

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

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).

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

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

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

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