Question 1

Solve for $x$ . Approximate the result to three decimal places. $$\log _ { 3 } x = \frac { 1 } { 2 }$$ 
A) $\sqrt { 3 } \approx - 1.732$
B) $\frac { \sqrt { 3 } } { 3 } \approx 1.732$
C) $\sqrt { 3 } \approx 1.732$
D) $\sqrt { 3 } = 1.732$
E) $\sqrt { 3 } \approx 2$

Accepted Answer

The equation $\log_3 x = \frac{1}{2}$ can be rewritten in exponential form as $3^{\frac{1}{2}} = x$, which simplifies to $x = \sqrt{3}$. The approximate value of $\sqrt{3}$ is 1.732.

Question 2

Write the logarithmic equation in exponential form. $$\log _ { 5 } 25 = 2$$ 
A) $5 ^ { 25 } = 2$
B) $5 ^ { 2 } = 25$
C) $5 ^ { 2 } = 10$
D) $25 ^ { 5 } = 2$
E) $5 ^ { 2 } = 55$

Accepted Answer

In exponential form, the base is raised to the exponent to equal the result of the logarithmic equation. Therefore, the exponential form of $$\log _ { 5 } 25 = 2$$ is $$5 ^ 2 = 25.$$ Choice B matches this format and is the correct answer.

Question 3

Select a scatter plot of the given data. 
A)

B)

C)

D)

E)

Accepted Answer

The answer of Select a scatter plot of the given...

Question 4

Evaluate the logarithm $\log _ { 1 / 3 } 1.681$ using the change of base formula. Round to 3 decimal places.
A) -0.473
B) 0.226
C) -0.571
D) 0.519
E) -2.115

Accepted Answer

The change of base formula allows us to convert $\log _ { 1 / 3 } 1.681$ into a more manageable form, such as $\log _ { 10 } 1.681 / \log _ { 10 } (1 / 3)$. Calculating this gives approximately -0.473.

Question 5

Solve the exponential equation algebraically. Approximate the result to three deci- mal places. $$e ^ { x } - 7 = 16$$ 
A) $\ln 23 \approx 3.135$
B) $\ln 23 \approx - 3.135$
C) $\ln 23 \approx - 2.773$
D) $\ln 23 \approx 2.773$
E) $\ln 23 \approx 1.946$

Accepted Answer

First, add 7 to both sides to get $e^x = 23$. Then, take the natural logarithm of both sides to solve for $x$, giving $x = \ln(23)$. Approximating $\ln(23)$ gives approximately 3.135.

Question 6

Evaluate the function at the indicated value of $x = a ^ { 8 }$ $g ( x ) = \log _ { a } x$ 
A) $- 8$
B) 8
C) 0
D) $\frac { 1 } { 8 }$
E) $- \frac { 1 } { 8 }$

Accepted Answer

We substitute the value of x into the function:

$$ g ( a ^ { 8 } ) = \log _ { a } a ^ { 8 } = 8 $$

Therefore, the answer is choice B.

Question 7

Use a graphing utility to construct a table of values for the function. Round your answer to two decimal places. 
$$f ( x ) = 3 ^ { - x }$$ 
A)
$\begin{array}{|c|c|c|c|c|c|}
\hline x & -2 & -1 & 0 & 1 & 2 \
\hline f(x) & -9 & 3 & 1 & 0.33 & 0.11 \
\hline
\end{array}$

B)
$\begin{array}{|c|c|c|c|c|c|}
\hline x & -2 & -1 & 0 & 1 & 2 \
\hline f(x) & 9 & 3 & 1 & 0.33 & -0.11 \
\hline
\end{array}$

C)
$\begin{array}{|c|c|c|c|c|c|}
\hline x & -2 & -1 & 0 & 1 & 2 \
\hline f(x) & 9 & -3 & 1 & 0.33 & 0.11 \
\hline
\end{array}$

D)
$\begin{array}{|c|c|c|c|c|c|}
\hline x & -2 & -1 & 0 & 1 & 2 \
\hline f(x) & 9 & 3 & 1 & 0.33 & 0.11 \
\hline
\end{array}$

E)
$\begin{array}{|c|c|c|c|c|c|}
\hline x & -2 & -1 & 0 & 1 & 2 \
\hline f(x) & 9 & 3 & 1 & -0.33 & 0.11 \
\hline
\end{array}$

Accepted Answer

By using a graphing utility, we can easily graph the function $f(x)=3^{-x}$:

[asy]
import TrigMacros;
size(200);
real ticklen=0.05cm;
real tickspace=0.5cm;
real xMin=-2;
real xMax=2;
real yMin=-2;
real yMax=10;
real function realpart (real x) {return 3^(-x);}
draw(graph(realpart,xMin,xMax),black+linewidth(1.5));
xaxis(xMin,xMax,Ticks(ticklen,tickspace));
yaxis(yMin,yMax,Ticks(ticklen,tickspace));
label("$f(x)=3^{-x}$",(0.7,4.3));
[/asy]

From the graph or by plugging in values for $x$ into the function, we can construct the table of values in choice $	extbf{(D)}$.

Question 8

Evaluate the function at the indicated value of $X$ . Round your result to three deci- mal places. $$\begin{array} { l c } 
\text { Function } & \text { Value } \
f ( x ) = 0.6 ^ { x } & x = 1.1
\end{array}$$ 
A) $0.57$
B) $- 1.1$
C) $1.1$
D) $0.6$
E) $- 0.57$

Accepted Answer

Plugging in $x=1.1$ into the function gives $f(1.1)=0.6^{1.1}\approx 0.57$. Rounded to three decimal places, the answer is A.

Question 9

Write the exponential equation in logarithmic form. $$3 ^ { - 2 } = \frac { 1 } { 9 }$$ 
A) $\log _ { 9 } \frac { 1 } { 3 } = - 2$
B) $\log _ { 3 } 9 = - 2$
C) $\log _ { 3 } 9 = 2$
D) $\log _ { 3 } \frac { 1 } { 9 } = 2$
E) $\log _ { 3 } \frac { 1 } { 9 } = - 2$

Accepted Answer

The logarithmic form of the exponential equation $3^{-2}=\frac{1}{9}$ is $\log_{3}\frac{1}{9}=-2$. Therefore, the correct choice is E. Choice A is incorrect because the base should be 3 instead of 9. Choice B is incorrect because it is the logarithmic form of $9^ {-2}$, not $3 ^ {- 2 }$. Choice C is incorrect because it gives the logarithmic form of $3 ^ { 2 }$, not $3 ^ { -2 }$. Choice D is incorrect because it gives the logarithmic form of $ \frac { 1 } { 3 }$ instead of $ \frac { 1 } { 9 }$.

Question 10

Evaluate the logarithm using the change-of-base formula. Round your result to three decimal places. $$\log _ { 4 } 6$$ 
A) $5.292$
B) $1.292$
C) $2.292$
D) $4.292$
E) $3.292$

Accepted Answer

The change-of-base formula allows us to rewrite the logarithm in terms of logarithms with a base of our choice, commonly 10 or e. Using base 10, we have $\log_4 6 = \frac{\log_{10} 6}{\log_{10} 4}$. Calculating this gives approximately 1.292.

Question 11

Condense the expression to the logarithm of a single quantity. $$\log x - 6 \log y + 5 \log z$$ 
A) $\log \frac { y ^ { 6 } } { x z ^ { 5 } }$
B) $\log \frac { x } { y ^ { 6 } z ^ { 5 } }$
C) $\log \frac { x y ^ { 6 } } { z ^ { 5 } }$
D) $\log \frac { x z ^ { 5 } } { y ^ { 6 } }$
E) $\log \frac { z ^ { 5 } } { x y ^ { 6 } }$

Accepted Answer

Using the properties of logarithms, we can condense the expression as follows:

\begin{align*}
\log x - 6 \log y + 5 \log z &= \log \frac{x}{y^6} + \log z^5 \
&= \log \frac{x}{y^6} \cdot z^5 \
&= \log \frac{x z^5}{y^6}.
\end{align*}

Therefore, the best choice is $\boxed{	extbf{(D)}}$.

Question 12

Find the exponential model that $y = a e ^ { b x }$ fits the points shown in the table. $$\begin{array} { | l | l | l | } 
\hline x & 0 & 4 \
\hline y & 25 & 5 \
\hline
\end{array}$$ 
A) $y = 25 e ^ { - 0.4024 x }$
B) $y = 25 e ^ { 0.5024 x }$
C) $y = - \frac { 1 } { 25 } e ^ { 0.4024 x }$
D) $y = \frac { 1 } { 25 } e ^ { - 0.6024 x }$
E) $y = - \frac { 1 } { 25 } e ^ { - 0.4024 x }$

Accepted Answer

Given the points (0, 25) and (4, 5), we can use the first point to find $a$ because when $x=0$, $y=a$. Thus, $a=25$. Using the second point (4, 5) and the value of $a$, we can solve for $b$ in the equation $5 = 25e^{4b}$. Simplifying gives $e^{4b} = \frac{1}{5}$, and solving for $b$ gives $b \approx -0.4024$. Therefore, the correct model is $y = 25e^{-0.4024x}$.

Question 13

Find the magnitude R of each earthquake of intensity I let I₀ $=1$. I=262300000 A) 7.42 B) 6.42 C) 8.42 D) 10.42 E) 9.42

Accepted Answer

The magnitude R of an earthquake is given by: 
$R = \frac{2}{3} \log_{10} \frac{I}{I_0}$ 
where I is the intensity of the earthquake and $I_0$ is a reference intensity of $1 \mu Pa$.
Using the given value of $I=262300000$, we can solve for R: 
$R = \frac{2}{3} \log_{10} \frac{262300000}{1} \approx 8.42$
Therefore, the answer is choice C).

Question 14

Find the exact value of the logarithmic expression without using a calculator. $$\log _ { 9 } \sqrt [ 5 ] { 9 }$$ 
A) 5
B) 45
C) 9
D) $\frac { 1 } { 5 }$
E) None of these

Accepted Answer

The expression can be simplified using the logarithm power rule, $\log_b (a^n) = n \log_b (a)$, where $b$ is the base of the logarithm, $a$ is the argument, and $n$ is the exponent. In this case, $\log_9 (\sqrt[5]{9})$ is equivalent to $\log_9 (9^{1/5})$, which simplifies to $\frac{1}{5} \log_9 (9)$. Since $\log_9 (9) = 1$, the expression simplifies to $\frac{1}{5}$.

Question 15

Rewrite the logarithm as a ratio of common logarithms. $$\log _ { 5 } 18$$ 
A) $\frac { \log 5 } { \log 18 }$
B) $\log \frac { 5 } { 18 }$
C) $\frac { \log 18 } { \log 5 }$
D) $\log \frac { 18 } { 5 }$
E) None of these

Accepted Answer

Use the change of base formula for logarithms, which states that $$\log _ { b } a = \frac { \log a } { \log b }$$
Applying this to the given problem, we get $$\log _ {5} 18 = \frac{\log 18}{\log 5}$$ Therefore, the answer is (C).

Question 16

Write the exponential equation in logarithmic form. $$e ^ { 9 x } = 10$$ 
A) $\ln 10 = - 9 x$
B) $\ln 9 = 10 x$
C) $\ln 10 = 9 x$
D)$\ln 10 = \frac { 1 } { 9 } x$
E)$\ln 10 = -\frac { 1 } { 9 } x$

Accepted Answer

Taking the natural logarithm of both sides, we have $$\ln(e^{9x}) = \ln(10)$$ $$9x \ln(e) = \ln(10)$$ Since $\ln(e) = 1$, we can simplify this to $$9x = \ln(10)$$ Dividing both sides by 9, we get $$x = \frac{1}{9} \ln(10)$$ Therefore, the correct choice is $\boxed{\textbf{(C)}}$.

Question 17

Select the graph of the exponential function. $$s ( t ) = 5 e ^ { 0.15 t }$$ 
A)

B)

C)

D)

E)

Accepted Answer

The answer of Select the graph of the exponential function....

Question 18

Evaluate the function at the indicated value of $x$ . Round your result to three decimal places. 
$ \begin{array}{ll}\text { Function } & \text { Value } \ h(x)=e^{x} & x=7/8\end{array} $
A) $0.417$
B) $0.875$
C) $8.07$
D) $7.5$
E) $1.398$

Accepted Answer

The answer of Evaluate the function at the indicated value...

Question 19

Use the One-to-One Property to solve the equation for x. $$\log _ { 5 } ( 3 x + 3 ) = \log _ { 5 } 10$$ 
A) $\frac { 1 } { 10 }$
B) $\frac { 2 } { 13 }$
C) $\frac { 2 } { 7 }$
D) $\frac { 7 } { 3 }$
E) $\frac { 13 } { 2 }$

Accepted Answer

Using the One-to-One Property, we can set the expressions inside the logarithms equal to each other:

$3x+3=10$

$3x=7$

$x=\frac{7}{3}$

Therefore, the answer is D.

Question 20

Determine whether the given x-value is a solution (or an approximate solution) of the equation. $$\begin{array} { l } 
4 ^ { 2 x - 7 } = 64 \
x = 5
\end{array}$$

Accepted Answer

Substitute x = 5 in the given equation:
$4^{2(5)-7} = 4^3 = 64$
Since the left side of the equation is equal to the right side when x = 5, 5 is a solution of the equation.

Quiz 3: Exponential and Logarithmic Functions

Solve for $x$ . Approximate the result to three decimal places. $\log _ { 3 } x = \frac { 1 } { 2 }$

Write the logarithmic equation in exponential form. $\log _ { 5 } 25 = 2$

Select a scatter plot of the given data.

Evaluate the logarithm $\log _ { 1 / 3 } 1.681$ using the change of base formula. Round to 3 decimal places.

Solve the exponential equation algebraically. Approximate the result to three deci- mal places. $e ^ { x } - 7 = 16$

Evaluate the function at the indicated value of $x = a ^ { 8 }$ $g ( x ) = \log _ { a } x$

Use a graphing utility to construct a table of values for the function. Round your answer to two decimal places. $f ( x ) = 3 ^ { - x }$

Evaluate the function at the indicated value of $X$ . Round your result to three deci- mal places. $\begin{array} { l c } \text { Function } & \text { Value } \\f ( x ) = 0.6 ^ { x } & x = 1.1\end{array}$

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

Evaluate the logarithm using the change-of-base formula. Round your result to three decimal places. $\log _ { 4 } 6$

Condense the expression to the logarithm of a single quantity. $\log x - 6 \log y + 5 \log z$

Find the exponential model that $y = a e ^ { b x }$ fits the points shown in the table. $\begin{array} { | l | l | l | } \hline x & 0 & 4 \\\hline y & 25 & 5 \\\hline\end{array}$

Find the magnitude R of each earthquake of intensity I let I₀ $=1$ . I=262300000

Find the exact value of the logarithmic expression without using a calculator. $\log _ { 9 } \sqrt [ 5 ] { 9 }$

Rewrite the logarithm as a ratio of common logarithms. $\log _ { 5 } 18$

Write the exponential equation in logarithmic form. $e ^ { 9 x } = 10$

Select the graph of the exponential function. $s ( t ) = 5 e ^ { 0.15 t }$

Evaluate the function at the indicated value of $x$ . Round your result to three decimal places. $\begin{array}{ll}\text { Function } & \text { Value } \\ h(x) =e^{x} & x=7/8\end{array}$

Use the One-to-One Property to solve the equation for x. $\log _ { 5 } ( 3 x + 3 ) = \log _ { 5 } 10$

Determine whether the given x-value is a solution (or an approximate solution) of the equation. $\begin{array} { l } 4 ^ { 2 x - 7 } = 64 \\x = 5\end{array}$

Quiz 3: Exponential and Logarithmic Functions

Solve for xxx . Approximate the result to three decimal places. log⁡3x=12\log _ { 3 } x = \frac { 1 } { 2 }log3​x=21​

Write the logarithmic equation in exponential form. log⁡525=2\log _ { 5 } 25 = 2log5​25=2

Select a scatter plot of the given data.

Evaluate the logarithm log⁡1/31.681\log _ { 1 / 3 } 1.681log1/3​1.681 using the change of base formula. Round to 3 decimal places.

Solve the exponential equation algebraically. Approximate the result to three deci- mal places. ex−7=16e ^ { x } - 7 = 16ex−7=16

Evaluate the function at the indicated value of x=a8x = a ^ { 8 }x=a8 g(x) =log⁡axg ( x ) = \log _ { a } xg(x) =loga​x

Use a graphing utility to construct a table of values for the function. Round your answer to two decimal places. f(x) =3−xf ( x ) = 3 ^ { - x }f(x) =3−x

Evaluate the function at the indicated value of XXX . Round your result to three deci- mal places. Function Value f(x) =0.6xx=1.1\begin{array} { l c } \text { Function } & \text { Value } \\f ( x ) = 0.6 ^ { x } & x = 1.1\end{array} Function f(x) =0.6x​ Value x=1.1​

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

Evaluate the logarithm using the change-of-base formula. Round your result to three decimal places. log⁡46\log _ { 4 } 6log4​6

Condense the expression to the logarithm of a single quantity. log⁡x−6log⁡y+5log⁡z\log x - 6 \log y + 5 \log zlogx−6logy+5logz

Find the exponential model that y=aebxy = a e ^ { b x }y=aebx fits the points shown in the table. x04y255\begin{array} { | l | l | l | } \hline x & 0 & 4 \\\hline y & 25 & 5 \\\hline\end{array}xy​025​45​​

Find the magnitude R of each earthquake of intensity I let I0 =1=1=1 . I=262300000

Find the exact value of the logarithmic expression without using a calculator. log⁡995\log _ { 9 } \sqrt [ 5 ] { 9 }log9​59​

Rewrite the logarithm as a ratio of common logarithms. log⁡518\log _ { 5 } 18log5​18

Write the exponential equation in logarithmic form. e9x=10e ^ { 9 x } = 10e9x=10

Select the graph of the exponential function. s(t) =5e0.15ts ( t ) = 5 e ^ { 0.15 t }s(t) =5e0.15t

Evaluate the function at the indicated value of xxx . Round your result to three decimal places. Function Value h(x) =exx=7/8 \begin{array}{ll}\text { Function } & \text { Value } \\ h(x) =e^{x} & x=7/8\end{array} Function h(x) =ex​ Value x=7/8​

Use the One-to-One Property to solve the equation for x. log⁡5(3x+3) =log⁡510\log _ { 5 } ( 3 x + 3 ) = \log _ { 5 } 10log5​(3x+3) =log5​10

Determine whether the given x-value is a solution (or an approximate solution) of the equation. 42x−7=64x=5\begin{array} { l } 4 ^ { 2 x - 7 } = 64 \\x = 5\end{array}42x−7=64x=5​