Question 1

Evaluate the function  f(x) =log⁡4xf ( x )  = \log _ { 4 } xf(x) =log4​x  at  x=116x = \frac { 1 } { 16 }x=161​  without using a calculator.

Accepted Answer

A)    −1- 1−1 
B)    −2- 2−2 
C)    −3- 3−3 
D)    161616 
E)    116\frac { 1 } { 16 }161​

Question 2

Simplify the expression below.  1+eln⁡x61 + e ^ { \ln x ^ { 6 } }1+elnx6

Accepted Answer

A)    1+e61 + e ^ { 6 }1+e6 
B)    1+x61 + x ^ { 6 }1+x6 
C)    1+ln⁡61 + \ln 61+ln6 
D)    ln⁡7\ln 7ln7 
E)    1+6ln⁡x1 + 6 \ln x1+6lnx

Question 3

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

Question 4

Condense the expression  15[log⁡4x+log⁡46]−[log⁡4y]\frac { 1 } { 5 } \left[ \log _ { 4 } x + \log _ { 4 } 6 \right] - \left[ \log _ { 4 } y \right]51​[log4​x+log4​6]−[log4​y]  to the logarithm of a single term.

Accepted Answer

A)    log⁡4(6x) 5y\log _ { 4 } \frac { ( 6 x )  ^ { 5 } } { y }log4​y(6x) 5​ 
B)    log⁡46x5y\log _ { 4 } \frac { 6 x } { 5 y }log4​5y6x​ 
C)    log⁡46xy5\log _ { 4 } \sqrt [ 5 ] { \frac { 6 x } { y } }log4​5y6x​l0 -0c4,-6.7,10,-10,18,-10 H400000v40H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5c53.7,-170.3,84.5,-266.8,92.5,-289.5zM1001 80h400000v40h-400000z'/>​ 
D)    log⁡46x5y\log _ { 4 } \frac { \sqrt [ 5 ] { 6 x } } { y }log4​y56xc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​​ 
E)    log⁡46x5−log⁡4y\log _ { 4 } \sqrt [ 5 ] { 6 x } - \log _ { 4 } ylog4​56xc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​−log4​y

Question 5

Solve the logarithmic equation below algebraically. Round your result to three decimal places.  5log⁡4(x+1) =125 \log _ { 4 } ( x + 1 )  = 125log4​(x+1) =12

Accepted Answer

A)   21.326
B)   22.244
C)   23.558
D)   23.291
E)   26.858

Question 6

Solve the logarithmic equation below algebraically.
 ln⁡(x+3) =ln⁡(x+1) −ln⁡(x+7) \ln ( x + 3 )  = \ln ( x + 1 )  - \ln ( x + 7 ) ln(x+3) =ln(x+1) −ln(x+7)

Accepted Answer

A)    111  and  666 
B)    −5- 5−5  and  −4- 4−4 
C)    −4- 4−4  and  111 
D)    −4- 4−4  and  666 
E)    −5- 5−5  and  111

Question 7

Evaluate the logarithm  log⁡1/30.211\log _ { 1 / 3 } 0.211log1/3​0.211  using the change of base formula. Round to 3 decimal places.

Accepted Answer

A)   -1.556
B)   1.709
C)   0.706
D)   1.416
E)   -0.676

Question 8

Solve the logarithmic equation below algebraically.
 ln⁡(x−6) =ln⁡(x+6) −ln⁡(x−4) \ln ( x - 6 )  = \ln ( x + 6 )  - \ln ( x - 4 ) ln(x−6) =ln(x+6) −ln(x−4)

Accepted Answer

A)   - 2 and 5
B)   2 and 9
C)   5 and 9
D)   - 2 and 9
E)   2 and 5

Question 9

Condense the expression below to the logarithm of a single quantity.  32log⁡7(z+5) \frac { 3 } { 2 } \log _ { 7 } ( z + 5 ) 23​log7​(z+5)

Accepted Answer

A)    log⁡7(z3+5) \log _ { 7 } ( \sqrt [ 3 ] { z } + 5 ) log7​(3zc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​+5)  
B)    log⁡7(z3+5) \log _ { 7 } \left( \sqrt { z ^ { 3 } } + 5 \right) log7​(z3c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​+5)  
C)    log⁡7z+53\log _ { 7 } \sqrt [ 3 ] { z + 5 }log7​3z+5c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​ 
D)    log⁡73(z+5) 2\log _ { 7 } \frac { 3 ( z + 5 )  } { 2 }log7​23(z+5) ​ 
E)    log⁡7(z+5) 3\log _ { 7 } \sqrt { ( z + 5 )  ^ { 3 } }log7​(z+5) 3c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120c340,-704.7,510.7,-1060.3,512,-1067l0 -0c4.7,-7.3,11,-11,19,-11H40000v40H1012.3s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60zM1001 80h400000v40h-400000z'/>​

Question 10

Evaluate the function  f(x) =14ln⁡x at x=14.67f ( x )  = \frac { 1 } { 4 } \ln x 	ext { at } x = 14.67f(x) =41​lnx at x=14.67  . Round to 3 decimal places. (You may use your calculator.)

Accepted Answer

A)   1.546
B)   0.671
C)   1.280
D)   -0.347
E)   undefined

Question 11

Identify the graph that represents the function.
 y=5e−2(x−1) 2y = 5 e ^ { - 2 ( x - 1 )  ^ { 2 } }y=5e−2(x−1) 2

Accepted Answer

A)  
B)  
C)  
D)  
E)

Question 12

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

Accepted Answer

A)  
B)  
C)   
D)  
E)

Question 13

Solve the logarithmic equation below algebraically. Round your result to three decimal places.
 ln⁡x+6=2\ln \sqrt { x + 6 } = 2lnx+6​=2

Accepted Answer

A)    60.26260.26260.262 
B)    97.68197.68197.681 
C)    65.96865.96865.968 
D)    67.92767.92767.927 
E)    48.59848.59848.598

Question 14

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) 
 log⁡4y2\log _ { 4 } \frac { y } { 2 }log4​2y​

Accepted Answer

A)    2log⁡4y2 \log _ { 4 } y2log4​y 
B)    y−log⁡42y - \log _ { 4 } 2y−log4​2 
C)    log⁡4y−2\log _ { 4 } y - 2log4​y−2 
D)    log⁡4y2\frac { \log _ { 4 } y } { 2 }2log4​y​ 
E)    log⁡4y−log⁡42\log _ { 4 } y - \log _ { 4 } 2log4​y−log4​2

Question 15

Solve the equation below for x. 
 log⁡557=x\log _ { 5 } 5 ^ { 7 } = xlog5​57=x

Accepted Answer

A)   8
B)   2
C)   5
D)   9
E)   7

Question 16

Solve  ln⁡x2−ln⁡3=0\ln x ^ { 2 } - \ln 3 = 0lnx2−ln3=0  for  xxx .

Accepted Answer

A)   9
B)    −3,3- \sqrt { 3 } , \sqrt { 3 }−3c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​,3c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​ 
C)    e9e ^ { 9 }e9 
D)    e3/2e ^ { 3 / 2 }e3/2 
E)   no solution

Question 17

Solve the exponential equation below algebraically. Round your result to three decimal places.  250[(1+0.03) x0.03]=160,000250 \left[ \frac { ( 1 + 0.03 )  ^ { x } } { 0.03 } \right] = 160,000250[0.03(1+0.03) x​]=160,000

Accepted Answer

A)   86.873
B)   99.967
C)   111.803
D)   97.718
E)   87.344

Question 18

Identify the graph that represents the function.
 y=ln⁡1x+1y = \ln \frac { 1 } { x + 1 }y=lnx+11​

Accepted Answer

A)  
B)  
C)  
D)  
E)

Question 19

Solve  (14) x=64\left( \frac { 1 } { 4 } \right)  ^ { x } = 64(41​) x=64  for x.

Accepted Answer

A)   1
B)   −1
C)   −3
D)   −4
E)   no solution

Question 20

What is the value of the function  f(x) =3.8e−1.5xf ( x )  = 3.8 e ^ { - 1.5 x }f(x) =3.8e−1.5x  at  x=2.5?x = 2.5 ?x=2.5?  Round to 3 decimal places.

Accepted Answer

A)    0.8480.8480.848 
B)    0.0890.0890.089 
C)    10.32910.32910.329 
D)    −38.736- 38.736−38.736 
E)    0.0010.0010.001

Exam 3: Exponential and Logarithmic Functions

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.

Correct AnswerverifiedShow Answer

Solve the equation below for x. log⁡557=x\log _ { 5 } 5 ^ { 7 } = xlog5​57=x

Correct AnswerverifiedShow Answer

Solve the logarithmic equation below algebraically. ln⁡(x+3) =ln⁡(x+1) −ln⁡(x+7) \ln ( x + 3 ) = \ln ( x + 1 ) - \ln ( x + 7 ) ln(x+3) =ln(x+1) −ln(x+7)

Correct AnswerverifiedShow Answer

Evaluate the function f(x) =log⁡4xf ( x ) = \log _ { 4 } xf(x) =log4​x at x=116x = \frac { 1 } { 16 }x=161​ without using a calculator.

Correct AnswerverifiedShow Answer

Evaluate the function f(x) =14ln⁡x at x=14.67f ( x ) = \frac { 1 } { 4 } \ln x \text { at } x = 14.67f(x) =41​lnx at x=14.67 . Round to 3 decimal places. (You may use your calculator.)

Correct AnswerverifiedShow Answer

Identify the graph that represents the function. y=5e−2(x−1) 2y = 5 e ^ { - 2 ( x - 1 ) ^ { 2 } }y=5e−2(x−1) 2

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

Solve ln⁡x2−ln⁡3=0\ln x ^ { 2 } - \ln 3 = 0lnx2−ln3=0 for xxx .

Correct AnswerverifiedShow Answer

Solve the logarithmic equation below algebraically. Round your result to three decimal places. ln⁡x+6=2\ln \sqrt { x + 6 } = 2lnx+6​=2

Correct AnswerverifiedShow Answer

Evaluate the logarithm log⁡1/30.211\log _ { 1 / 3 } 0.211log1/3​0.211 using the change of base formula. Round to 3 decimal places.

Correct AnswerverifiedShow Answer

Solve the exponential equation below algebraically. Round your result to three decimal places. 250[(1+0.03) x0.03]=160,000250 \left[ \frac { ( 1 + 0.03 ) ^ { x } } { 0.03 } \right] = 160,000250[0.03(1+0.03) x​]=160,000

Correct AnswerverifiedShow Answer

Solve the logarithmic equation below algebraically. Round your result to three decimal places. 5log⁡4(x+1) =125 \log _ { 4 } ( x + 1 ) = 125log4​(x+1) =12

Correct AnswerverifiedShow Answer

Identify the graph that represents the function. y=ln⁡1x+1y = \ln \frac { 1 } { x + 1 }y=lnx+11​

Correct AnswerverifiedShow Answer

Solve (14) x=64\left( \frac { 1 } { 4 } \right) ^ { x } = 64(41​) x=64 for x.

Correct AnswerverifiedShow Answer

Condense the expression below to the logarithm of a single quantity. 32log⁡7(z+5) \frac { 3 } { 2 } \log _ { 7 } ( z + 5 ) 23​log7​(z+5)

Correct AnswerverifiedShow Answer

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) log⁡4y2\log _ { 4 } \frac { y } { 2 }log4​2y​

Correct AnswerverifiedShow Answer

What is the value of the function f(x) =3.8e−1.5xf ( x ) = 3.8 e ^ { - 1.5 x }f(x) =3.8e−1.5x at x=2.5?x = 2.5 ?x=2.5? Round to 3 decimal places.

Correct AnswerverifiedShow Answer

Solve the logarithmic equation below algebraically. ln⁡(x−6) =ln⁡(x+6) −ln⁡(x−4) \ln ( x - 6 ) = \ln ( x + 6 ) - \ln ( x - 4 ) ln(x−6) =ln(x+6) −ln(x−4)

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Condense the expression 15[log⁡4x+log⁡46]−[log⁡4y]\frac { 1 } { 5 } \left[ \log _ { 4 } x + \log _ { 4 } 6 \right] - \left[ \log _ { 4 } y \right]51​[log4​x+log4​6]−[log4​y] to the logarithm of a single term.

Correct AnswerverifiedShow Answer

Simplify the expression below. 1+eln⁡x61 + e ^ { \ln x ^ { 6 } }1+elnx6

Correct AnswerverifiedShow Answer

Correct Answer
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Solve the equation below for x. $\log _ { 5 } 5 ^ { 7 } = x$

Correct Answer
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Solve the logarithmic equation below algebraically. $\ln ( x + 3 ) = \ln ( x + 1 ) - \ln ( x + 7 )$

Correct Answer
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Evaluate the function $f ( x ) = \log _ { 4 } x$ at $x = \frac { 1 } { 16 }$ without using a calculator.

Correct Answer
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Evaluate the function $f ( x ) = \frac { 1 } { 4 } \ln x \text { at } x = 14.67$ . Round to 3 decimal places. (You may use your calculator.)

Correct Answer
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Identify the graph that represents the function. $y = 5 e ^ { - 2 ( x - 1 ) ^ { 2 } }$

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

Correct Answer
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Solve $\ln x ^ { 2 } - \ln 3 = 0$ for $x$ .

Correct Answer
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Solve the logarithmic equation below algebraically. Round your result to three decimal places. $\ln \sqrt { x + 6 } = 2$

Correct Answer
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Evaluate the logarithm $\log _ { 1 / 3 } 0.211$ using the change of base formula. Round to 3 decimal places.

Correct Answer
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Solve the exponential equation below algebraically. Round your result to three decimal places. $250 \left[ \frac { ( 1 + 0.03 ) ^ { x } } { 0.03 } \right] = 160,000$

Correct Answer
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Solve the logarithmic equation below algebraically. Round your result to three decimal places. $5 \log _ { 4 } ( x + 1 ) = 12$

Correct Answer
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Identify the graph that represents the function. $y = \ln \frac { 1 } { x + 1 }$

Correct Answer
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Solve $\left( \frac { 1 } { 4 } \right) ^ { x } = 64$ for x.

Correct Answer
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Condense the expression below to the logarithm of a single quantity. $\frac { 3 } { 2 } \log _ { 7 } ( z + 5 )$

Correct Answer
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Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) $\log _ { 4 } \frac { y } { 2 }$

Correct Answer
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What is the value of the function $f ( x ) = 3.8 e ^ { - 1.5 x }$ at $x = 2.5 ?$ Round to 3 decimal places.

Correct Answer
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Solve the logarithmic equation below algebraically. $\ln ( x - 6 ) = \ln ( x + 6 ) - \ln ( x - 4 )$

Correct Answer
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Condense the expression $\frac { 1 } { 5 } \left[ \log _ { 4 } x + \log _ { 4 } 6 \right] - \left[ \log _ { 4 } y \right]$ to the logarithm of a single term.

Correct Answer
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Simplify the expression below. $1 + e ^ { \ln x ^ { 6 } }$

Correct Answer
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