Question 1

Solve the equation: $81 ^ { 2 x } \cdot \left( \frac { 1 } { 9 } \right) ^ { x + 2 } = 729 \cdot \left( 9 ^ { x } \right) ^ { - 2 }$

Accepted Answer

A)  $x = 0$ 
B)  $x = 2$ 
C)  $x = 1$ 
D)  $x = 1,2$ 
E)  $x = 0,3$ 
A)  $x = 0$ 
B)  $x = 2$ 
C)  $x = 1$ 
D)  $x = 1,2$ 
E)  $x = 0,3$ C

Question 2

Write the expression as one logarithm. 9ln ( x ) - 7ln $\left( \frac { 1 } { y } \right)$ - 4ln ( xy )

Accepted Answer

A)  ln ( x 9  - y 7  - ( xy )  4  ) 
B)  ln ( 9x - 7y - 4xy ) 
C)  ln ( x 3 y 5  ) 
D)  ln ( x 5 y 3  ) 
A)  ln ( x 9  - y 7  - ( xy )  4  ) 
B)  ln ( 9x - 7y - 4xy ) 
C)  ln ( x 3 y 5  ) 
D)  ln ( x 5 y 3  ) D

Question 3

Use the theorem on inverse functions to determine whether f and g are inverse functions of each other. $f ( x ) = 6 x - 1 ; g ( x ) = \frac { x + 1 } { 6 }$

Accepted Answer

A)  f and g are not inverse functions of each other. 
B)  f and g are inverse functions of each other. 
A)  f and g are not inverse functions of each other. 
B)  f and g are inverse functions of each other. B

Question 4

Express in terms of logarithms of x, y, z, or w. $\log _ { 7 } \frac { x ^ { 4 } w } { y ^ { 2 } z ^ { 3 } }$

Accepted Answer

A)  4log  7  ( x ) + log  7  ( w ) - 3log  7  ( y ) - 2log  7  ( z ) 
B)  4log  7  ( x ) + log  7  ( w ) + 2log  7  ( y ) + 3log  7  ( z ) 
C)  log  7  ( x ) + log  7  ( w ) - log  7  ( y ) + log  7  ( z ) 
D)  4log  7  ( x ) + log  7  ( w ) - 2log  7  ( y ) - 3log  7  ( z ) 
E)  4log  7  ( x ) - log  7  ( w ) + 2log  7  ( y ) - 3log  7  ( z ) 
A)  4log  7  ( x ) + log  7  ( w ) - 3log  7  ( y ) - 2log  7  ( z ) 
B)  4log  7  ( x ) + log  7  ( w ) + 2log  7  ( y ) + 3log  7  ( z ) 
C)  log  7  ( x ) + log  7  ( w ) - log  7  ( y ) + log  7  ( z ) 
D)  4log  7  ( x ) + log  7  ( w ) - 2log  7  ( y ) - 3log  7  ( z ) 
E)  4log  7  ( x ) - log  7  ( w ) + 2log  7  ( y ) - 3log  7  ( z )

Question 5

If $900 is deposited in a savings account that pays interest at a rate of $6 \frac { 1 } { 4 }$ % per year compounded continuously, find the balance after 7 years.

Accepted Answer

A)  $1,418.56 
B)  $581.08 
C)  $1,369.77 
D)  $1,393.95 
E)  $3,484.87 
A)  $1,418.56 
B)  $581.08 
C)  $1,369.77 
D)  $1,393.95 
E)  $3,484.87

Question 6

Solve the equation. $\ln x = 1 - \ln ( x + 4 )$

Accepted Answer

A)  $x = - 2 - \sqrt { 4 + \mathrm { e } }$ 
B)  $x = 4 - \sqrt { 16 + \mathrm { e } }$ 
C)  $x = - 4 + \sqrt { 16 + \mathrm { e } }$ 
D)  $x = - 4 - \sqrt { 16 + \mathrm { e } }$ 
E)  $x = - 2 + \sqrt { 4 + \mathrm { e } }$ 
A)  $x = - 2 - \sqrt { 4 + \mathrm { e } }$ 
B)  $x = 4 - \sqrt { 16 + \mathrm { e } }$ 
C)  $x = - 4 + \sqrt { 16 + \mathrm { e } }$ 
D)  $x = - 4 - \sqrt { 16 + \mathrm { e } }$ 
E)  $x = - 2 + \sqrt { 4 + \mathrm { e } }$

Question 7

The air pressure p (h) (in lb/ in 2 ) at an altitude of h feet above sea level may be approximated by the formula $p ( h ) = 14.7 e ^ { - 0.0000385 h }$ At approximately what altitude h is the air pressure 9 lb/ in 2  ?

Accepted Answer

A)  12,743 ft 
B)  1,274 ft 
C)  17,123 ft 
D)  5,534 ft 
E)  553 ft 
A)  12,743 ft 
B)  1,274 ft 
C)  17,123 ft 
D)  5,534 ft 
E)  553 ft

Question 8

Sketch the graph of $$f ( x ) = \log _ { 3 } \left( x ^ { 2 } \right)$$

Accepted Answer

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

Question 9

Find the solution of the equation. $4 ^ { - x } = 64$

Accepted Answer

A)  $x = \frac { 1 } { 2 }$ 
B)  $x = \frac { 1 } { 3 }$ 
C)  $x = - 3$ 
D)  $x = - 5$ 
E)  $x = 2$ 
A)  $x = \frac { 1 } { 2 }$ 
B)  $x = \frac { 1 } { 3 }$ 
C)  $x = - 3$ 
D)  $x = - 5$ 
E)  $x = 2$

Question 10

How much money, invested at an interest rate of 11% per year compounded continuously, will amount to $105,000 after 15 years?

Accepted Answer

A)  $20,165.24 
B)  $117,209.20 
C)  $5,467.33 
D)  $40,330.48 
E)  $104,983.26 
A)  $20,165.24 
B)  $117,209.20 
C)  $5,467.33 
D)  $40,330.48 
E)  $104,983.26

Question 11

The radioactive bismuth isotope  210 Bi has a half-life of 5 days. If there is 100 milligrams of 210 Bi present at t = 0, then the amount f(t) remaining after t days is given by: $f ( t ) = 100 ( 2 ) ^ { - \frac { t } { 5 } }$ How much 210 Bi remains after 25 days?

Accepted Answer

A)  3.125 milligrams 
B)  12.125 milligrams 
C)  5.125 milligrams 
D)  8.125 milligrams 
E)  2.125 milligrams 
A)  3.125 milligrams 
B)  12.125 milligrams 
C)  5.125 milligrams 
D)  8.125 milligrams 
E)  2.125 milligrams

Question 12

Find an exponential function of the form f (x) = ba x  that has the given y-intercept and passes through the point P: y-intercept = 125; P (3,1);

Accepted Answer

A)  $y = 5 \left( \frac { 1 } { 125 } \right) ^ { x }$ 
B)  $y = \frac { 1 } { 125 } \left( \frac { 1 } { 5 } \right) ^ { x }$ 
C)  $$y = 125 \left( \frac { 1 } { 5 } \right) ^ { x }$$ 
D)  $y = 5 \left( \frac { 1 } { 5 } \right) ^ { x }$ 
E)  $y = 125 \left( \frac { 1 } { 125 } \right) ^ { x }$ 
A)  $y = 5 \left( \frac { 1 } { 125 } \right) ^ { x }$ 
B)  $y = \frac { 1 } { 125 } \left( \frac { 1 } { 5 } \right) ^ { x }$ 
C)  $$y = 125 \left( \frac { 1 } { 5 } \right) ^ { x }$$ 
D)  $y = 5 \left( \frac { 1 } { 5 } \right) ^ { x }$ 
E)  $y = 125 \left( \frac { 1 } { 125 } \right) ^ { x }$

Question 13

Some lending institutions calculate the monthly payment M on a loan of L dollars at an interest rate r (expressed as a decimal) by using the formula: $M = \frac { L r k } { 12 ( k - 1 ) }$ where k = [1 + ( r / 12 )] 12t  and t is the number of years that the loan is in effect. An automobile dealer offers customers no-down-payment 8-year loans at an interest rate of 10%. If a customer can afford to pay $122 per month, find the price of the most expensive car that can be purchased.

Accepted Answer

A)  $ 8,050.00 
B)  $ 8,045.62 
C)  $ 8,039.98 
D)  $ 8,048.79 
E)  $ 8,029.63 
A)  $ 8,050.00 
B)  $ 8,045.62 
C)  $ 8,039.98 
D)  $ 8,048.79 
E)  $ 8,029.63

Question 14

Sketch the graph of $f ( x ) = \log _ { 5 } \sqrt { x }$

Accepted Answer

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

Question 15

In 1980, the population of blue whales in the southern hemisphere was thought to number 5,000. The population N(t) has been decreasing according to the formula $N ( t ) = 5,000 e ^ { - 0.1485 t }$ where t is in years and t = 0 corresponds to 1980. Predict the population in the year 2010 if this trend continues.

Accepted Answer

A)  $3,203$ 
B)  $50$ 
C)  $58$ 
D)  $4,942$ 
E)  $1,797$ 
A)  $3,203$ 
B)  $50$ 
C)  $58$ 
D)  $4,942$ 
E)  $1,797$

Question 16

Find the inverse function of $f$ . $f ( x ) = 3 x + 6$

Accepted Answer

A)  $f ^ { - 1 } ( x ) = \frac { ( x + 6 ) } { 3 }$ 
B)  $f ^ { - 1 } ( x ) = \frac { ( x - 6 ) } { 3 }$ 
C)  $f ^ { - 1 } ( x ) = \frac { 3 } { ( x - 6 ) }$ 
D)  $f ^ { - 1 } ( x ) = \frac { 1 } { 3 } x - 6$ 
E)  $f ^ { - 1 } ( x ) = \frac { 3 } { ( x + 6 ) }$ 
A)  $f ^ { - 1 } ( x ) = \frac { ( x + 6 ) } { 3 }$ 
B)  $f ^ { - 1 } ( x ) = \frac { ( x - 6 ) } { 3 }$ 
C)  $f ^ { - 1 } ( x ) = \frac { 3 } { ( x - 6 ) }$ 
D)  $f ^ { - 1 } ( x ) = \frac { 1 } { 3 } x - 6$ 
E)  $f ^ { - 1 } ( x ) = \frac { 3 } { ( x + 6 ) }$

Question 17

An economist suspects that the points $( 0,6 ) , ( 1,3.152 ) , ( 2,1.656 ) , ( 3,0.870 )$ lie on the graph of $y = c 5 ^ { k x }$ , where $c$ and k are constants. If the data points have three-decimal-place accuracy, is this suspicion correct?

Accepted Answer

A)  The suspicion is incorrect 
B)  The suspicion is correct 
A)  The suspicion is incorrect 
B)  The suspicion is correct

Question 18

When the volume control on a stereo system is increased, the voltage across a loudspeaker changes from V 1  to V 2 , and the decibel increase is given by $d b = 20 \log \frac { V _ { 2 } } { V _ { 1 } }$ Find the decibel increase if the voltage changes from 1.7 volts to 3 volts.

Accepted Answer

A)  db = 4.9 
B)  db = 8.7 
C)  db = 7.2 
D)  db = 11.4 
E)  db = 3.8 
A)  db = 4.9 
B)  db = 8.7 
C)  db = 7.2 
D)  db = 11.4 
E)  db = 3.8

Question 19

The graph of a one-to-one function f is shown. Find the domain D and range R of the inverse function $f ^ { - 1 }$ .

Accepted Answer

A)  $D = ( - \infty , \infty ) , R = [ 2,8 ]$ 
B)  $D = [ 6,8 ] , R = [ 0,2 ]$ 
C)  $D = [ 0,6 ] , R = [ 2,8 ]$ 
D)  $D = [ 2,6 ] , R = [ 0,8 ]$ 
E)  $D = [ 0,6 ] , R = ( - \infty , \infty )$ 
A)  $D = ( - \infty , \infty ) , R = [ 2,8 ]$ 
B)  $D = [ 6,8 ] , R = [ 0,2 ]$ 
C)  $D = [ 0,6 ] , R = [ 2,8 ]$ 
D)  $D = [ 2,6 ] , R = [ 0,8 ]$ 
E)  $D = [ 0,6 ] , R = ( - \infty , \infty )$

Question 20

The graph of a one-to-one function f is shown. Use the reflection property to sketch the graph of $f ^ { - 1 }$

Accepted Answer

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

Solve the equation: $81 ^ { 2 x } \cdot \left( \frac { 1 } { 9 } \right) ^ { x + 2 } = 729 \cdot \left( 9 ^ { x } \right) ^ { - 2 }$

Write the expression as one logarithm. 9ln ( x ) - 7ln $\left( \frac { 1 } { y } \right)$ - 4ln ( xy )

Use the theorem on inverse functions to determine whether f and g are inverse functions of each other. $f ( x ) = 6 x - 1 ; g ( x ) = \frac { x + 1 } { 6 }$

Express in terms of logarithms of x, y, z, or w. $\log _ { 7 } \frac { x ^ { 4 } w } { y ^ { 2 } z ^ { 3 } }$

If $900 is deposited in a savings account that pays interest at a rate of $6 \frac { 1 } { 4 }$ % per year compounded continuously, find the balance after 7 years.

Solve the equation. $\ln x = 1 - \ln ( x + 4 )$

The air pressure p (h) (in lb/ in²) at an altitude of h feet above sea level may be approximated by the formula $p ( h ) = 14.7 e ^ { - 0.0000385 h }$ At approximately what altitude h is the air pressure 9 lb/ in² ?

Sketch the graph of $f ( x ) = \log _ { 3 } \left( x ^ { 2 } \right)$

Find the solution of the equation. $4 ^ { - x } = 64$

How much money, invested at an interest rate of 11% per year compounded continuously, will amount to $105,000 after 15 years?

The radioactive bismuth isotope ²¹⁰Bi has a half-life of 5 days. If there is 100 milligrams of ²¹⁰Bi present at t = 0, then the amount f(t) remaining after t days is given by: $f ( t ) = 100 ( 2 ) ^ { - \frac { t } { 5 } }$ How much ²¹⁰Bi remains after 25 days?

Find an exponential function of the form f (x) = ba^x that has the given y-intercept and passes through the point P: y-intercept = 125; P (3,1);

Sketch the graph of $f ( x ) = \log _ { 5 } \sqrt { x }$

Find the inverse function of $f$ . $f ( x ) = 3 x + 6$

An economist suspects that the points $( 0,6 ) , ( 1,3.152 ) , ( 2,1.656 ) , ( 3,0.870 )$ lie on the graph of $y = c 5 ^ { k x }$ , where $c$ and k are constants. If the data points have three-decimal-place accuracy, is this suspicion correct?

When the volume control on a stereo system is increased, the voltage across a loudspeaker changes from V₁ to V₂, and the decibel increase is given by $d b = 20 \log \frac { V _ { 2 } } { V _ { 1 } }$ Find the decibel increase if the voltage changes from 1.7 volts to 3 volts.

The graph of a one-to-one function f is shown. Find the domain D and range R of the inverse function $f ^ { - 1 }$ . $The graph of a one-to-one function f is shown. Find the domain D and range R of the inverse function f ^ { - 1 } .$

The graph of a one-to-one function f is shown. Use the reflection property to sketch the graph of $f ^ { - 1 }$ $The graph of a one-to-one function f is shown. Use the reflection property to sketch the graph of f ^ { - 1 }$

Topics From Algebra

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Exam 4: Inverse, Exponential, and Logarithmic Functions

Solve the equation: 812x⋅(19)x+2=729⋅(9x)−281 ^ { 2 x } \cdot \left( \frac { 1 } { 9 } \right) ^ { x + 2 } = 729 \cdot \left( 9 ^ { x } \right) ^ { - 2 }812x⋅(91​)x+2=729⋅(9x)−2

Write the expression as one logarithm. 9ln ( x ) - 7ln (1y)\left( \frac { 1 } { y } \right)(y1​) - 4ln ( xy )

Use the theorem on inverse functions to determine whether f and g are inverse functions of each other. f(x)=6x−1;g(x)=x+16f ( x ) = 6 x - 1 ; g ( x ) = \frac { x + 1 } { 6 }f(x)=6x−1;g(x)=6x+1​

Express in terms of logarithms of x, y, z, or w. log⁡7x4wy2z3\log _ { 7 } \frac { x ^ { 4 } w } { y ^ { 2 } z ^ { 3 } }log7​y2z3x4w​

If $900 is deposited in a savings account that pays interest at a rate of 6146 \frac { 1 } { 4 }641​ % per year compounded continuously, find the balance after 7 years.

Solve the equation. ln⁡x=1−ln⁡(x+4)\ln x = 1 - \ln ( x + 4 )lnx=1−ln(x+4)

The air pressure p (h) (in lb/ in 2 ) at an altitude of h feet above sea level may be approximated by the formula p(h)=14.7e−0.0000385hp ( h ) = 14.7 e ^ { - 0.0000385 h }p(h)=14.7e−0.0000385h At approximately what altitude h is the air pressure 9 lb/ in 2 ?

Sketch the graph of f(x)=log⁡3(x2)f ( x ) = \log _ { 3 } \left( x ^ { 2 } \right)f(x)=log3​(x2)

Find the solution of the equation. 4−x=644 ^ { - x } = 644−x=64

How much money, invested at an interest rate of 11% per year compounded continuously, will amount to $105,000 after 15 years?

The radioactive bismuth isotope 210 Bi has a half-life of 5 days. If there is 100 milligrams of 210 Bi present at t = 0, then the amount f(t) remaining after t days is given by: f(t)=100(2)−t5f ( t ) = 100 ( 2 ) ^ { - \frac { t } { 5 } }f(t)=100(2)−5t​ How much 210 Bi remains after 25 days?

Find an exponential function of the form f (x) = ba x that has the given y-intercept and passes through the point P: y-intercept = 125; P (3,1);

Sketch the graph of f(x)=log⁡5xf ( x ) = \log _ { 5 } \sqrt { x }f(x)=log5​x​

Find the inverse function of fff . f(x)=3x+6f ( x ) = 3 x + 6f(x)=3x+6

The graph of a one-to-one function f is shown. Find the domain D and range R of the inverse function f−1f ^ { - 1 }f−1 .

The graph of a one-to-one function f is shown. Use the reflection property to sketch the graph of f−1f ^ { - 1 }f−1

Topics From Algebra

Functions and Graphs

Polynomial and Rational Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometry

Systems of Equations and Inequalities

Sequences, Series, and Probability

Topics From Analytical Geometry

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Solve the equation: $81 ^ { 2 x } \cdot \left( \frac { 1 } { 9 } \right) ^ { x + 2 } = 729 \cdot \left( 9 ^ { x } \right) ^ { - 2 }$

Write the expression as one logarithm. 9ln ( x ) - 7ln $\left( \frac { 1 } { y } \right)$ - 4ln ( xy )

Use the theorem on inverse functions to determine whether f and g are inverse functions of each other. $f ( x ) = 6 x - 1 ; g ( x ) = \frac { x + 1 } { 6 }$

Express in terms of logarithms of x, y, z, or w. $\log _ { 7 } \frac { x ^ { 4 } w } { y ^ { 2 } z ^ { 3 } }$

If $900 is deposited in a savings account that pays interest at a rate of $6 \frac { 1 } { 4 }$ % per year compounded continuously, find the balance after 7 years.

Solve the equation. $\ln x = 1 - \ln ( x + 4 )$

The air pressure p (h) (in lb/ in²) at an altitude of h feet above sea level may be approximated by the formula $p ( h ) = 14.7 e ^ { - 0.0000385 h }$ At approximately what altitude h is the air pressure 9 lb/ in² ?

Sketch the graph of $f ( x ) = \log _ { 3 } \left( x ^ { 2 } \right)$

Find the solution of the equation. $4 ^ { - x } = 64$

The radioactive bismuth isotope ²¹⁰Bi has a half-life of 5 days. If there is 100 milligrams of ²¹⁰Bi present at t = 0, then the amount f(t) remaining after t days is given by: $f ( t ) = 100 ( 2 ) ^ { - \frac { t } { 5 } }$ How much ²¹⁰Bi remains after 25 days?

Find an exponential function of the form f (x) = ba^x that has the given y-intercept and passes through the point P: y-intercept = 125; P (3,1);

Sketch the graph of $f ( x ) = \log _ { 5 } \sqrt { x }$

Find the inverse function of $f$ . $f ( x ) = 3 x + 6$

The graph of a one-to-one function f is shown. Find the domain D and range R of the inverse function $f ^ { - 1 }$ . $The graph of a one-to-one function f is shown. Find the domain D and range R of the inverse function f ^ { - 1 } .$

The graph of a one-to-one function f is shown. Use the reflection property to sketch the graph of $f ^ { - 1 }$ $The graph of a one-to-one function f is shown. Use the reflection property to sketch the graph of f ^ { - 1 }$