Question 1

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

Accepted Answer

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

Question 2

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. $A = \$ 100,000$

Accepted Answer

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)  $P \approx \$ 19,691.17$ 
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 3

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

Accepted Answer

A)  $\frac { x - 2 } { 2 }$ 
B)  $\frac { x + 2 } { - 2 }$ 
C)  $\frac { x - 2 } { - 2 }$ 
D)  $\frac { - ( x + 2 ) } { 2 }$ 
E)  $\frac { x + 2 } { 2 }$ 
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 4

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

Accepted Answer

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)  $f ^ { - 1 } ( x ) = x + 5$ , $f \left( f ^ { - 1 } ( x ) \right) = x$ , $f ^ { - 1 } ( f ( x ) ) = x ^ { - 1 }$ 
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 5

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

Accepted Answer

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

Question 6

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

Accepted Answer

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

Question 7

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

Accepted Answer

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

Question 8

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

Accepted Answer

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

Question 9

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

Accepted Answer

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)  $6 \ln z - \frac { 4 } { 5 } \ln x + \frac { 4 } { 5 } \ln y$ 
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 10

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. $A = \$ 1,000,000 , r = 8 \% , t = 20 \text { years }, n = 2$

Accepted Answer

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)  $P \approx \$ 1008,289.04$ 
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 11

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

Accepted Answer

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

Question 12

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

Accepted Answer

A)  $x = - \frac { 5 } { 9 }$ 
B)  $x = \frac { 1 } { 9 }$ 
C)  $x = \frac { 1 } { 6 }$ 
D)  $x = - \frac { 5 } { 6 }$ 
E)  $x = \frac { 5 } { 6 }$ 
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 13

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

Accepted Answer

A)  $\log _ { 2 } 9 = 3$ 
B)  $\log _ { 3 } 2 = 9$ 
C)  $\log _ { 9 } 3 = 2$ 
D)  $\log _ { 2 } 3 = 9$ 
E)  $\log _ { 3 } 9 = 2$ 
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 14

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 $$x.$$ $$\begin{array} { | c | c | } 
\hline \text { Length, } x & \text { Diameter, } y \
\hline 1 & 0.095 \
\hline 2 & 0.156 \
\hline 3 & 0.208 \
\hline
\end{array}$$ $$\begin{array} { | c | c | } 
\hline \text { Length, } x & \text { Diameter, } y \
\hline 4 & 0.256 \
\hline 5 & 0.300 \
\hline 6 & 0.342 \
\hline
\end{array}$$

Accepted Answer

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)  $\ln y = 0.715 \ln x + \ln 0.095$ 
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 15

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

Accepted Answer

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

Question 16

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

Accepted Answer

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, $f ^ { - 1 } ( x ) = - 2 + x , x \geq 0$ 
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 17

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

Accepted Answer

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)  $f ^ { - 1 } = \{ ( 4,1 ) , ( 5,2 ) , ( 6,3 ) , ( 7,7 ) \}$ 
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 18

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.

Accepted Answer

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)  $\log _ { 5 }  \sqrt[5] { 6 x } - \log _ { 5 } y$ 
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 19

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

Accepted Answer

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)  $x = 1.386,4.000$ 
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 20

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

Accepted Answer

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

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

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. $A = \$ 100,000$

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

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

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

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

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

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

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

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

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

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

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 $x.$ Length, x Diameter, y 1 0.095 2 0.156 3 0.208 Length, x Diameter, y 4 0.256 5 0.300 6 0.342

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

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

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

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.

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

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

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Exam 5: Exponential and Logarithmic Functions

Find the vertical asymptote of the logarithmic function below. f(x)=log⁡2(x−4)f ( x ) = \log _ { 2 } ( x - 4 )f(x)=log2​(x−4)

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

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

Find the inverse function informally f(x)=x−5f ( x ) = x - 5f(x)=x−5 . 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

Use algebraic procedures to find the exact solution of the equation ln⁡x=12ln⁡(2x+199)+12ln⁡9\ln x = \frac { 1 } { 2 } \ln \left( 2 x + \frac { 19 } { 9 } \right) + \frac { 1 } { 2 } \ln 9lnx=21​ln(2x+919​)+21​ln9 .

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

Use the functions given by f(x)=18x−3f ( x ) = \frac { 1 } { 8 } x - 3f(x)=81​x−3 and g(x)=x2g ( x ) = x ^ { 2 }g(x)=x2 to find the value (f−1∘g−1)(1)\left( f ^ { - 1 } \circ g ^ { - 1 } \right) ( 1 )(f−1∘g−1)(1)

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

Expand the logarithmic expression ln⁡(z6x4y45)\ln \left( \frac { z ^ { 6 } } { \sqrt [ 5 ] { x ^ { 4 } y ^ { 4 } } } \right)ln(5x4y4​z6​) . Assume all variable expressions represent positive real numbers.

Write the logarithmic equation 4=log⁡2164 = \log _ { 2 } 164=log2​16 in its exponential form.

Use algebraic procedures to find the exact solution(s) of the equation below. 69x+2=12166 ^ { 9 x + 2 } = \frac { 1 } { 216 }69x+2=2161​

Write the exponential equation 32=93 ^ { 2 } = 932=9 in its logarithmic form.

The approximate lengths and diameters (in inches) of common nails are shown in the table. Find a logarithmic equation that relates the diameter yyy of a common nail to its length x.x.x. Length, x Diameter, y 1 0.095 2 0.156 3 0.208 Length, x Diameter, y 4 0.256 5 0.300 6 0.342

Solve for x. 23x=1282 ^ { 3 x } = 12823x=128

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

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

Condense the expression 15[log⁡5x+log⁡56]−[log⁡5y]\frac { 1 } { 5 } \left[ \log _ { 5 } x + \log _ { 5 } 6 \right] - \left[ \log _ { 5 } y \right]51​[log5​x+log5​6]−[log5​y] to the logarithm of a single term.

Solve the exponential equation algebraically. Approximate the result to three decimal places. e2x−5ex+4=0e ^ { 2 x } - 5 e ^ { x } + 4 = 0e2x−5ex+4=0

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

Fundamental Concepts of Algebra

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

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. $A = \$ 100,000$

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

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

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

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

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

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

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

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

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

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

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 $x.$ Length, x Diameter, y 1 0.095 2 0.156 3 0.208 Length, x Diameter, y 4 0.256 5 0.300 6 0.342

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

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

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

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.

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

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