Question 1

Solve the exponential equation algebraically. Approximate the result to three decimal places. 
$e ^ { 2 x } - 5 e ^ { x } - 6 = 0$ 
A)$\ln 6 \approx 3.792$
B) $\ln 6 \approx 2.792$
C) $\ln 6 \approx 0.792$
D) $\ln 6 \approx - 0.208$
E) $\ln 6 \approx 1.792$

Accepted Answer

E

Question 2

Write the exponential equation in logarithmic form. $4 ^ { - 2 } = \frac { 1 } { 16 }$ 
A) $\log _ { 16 } \frac { 1 } { 4 } = - 2$
B) $\log _ { 4 } \frac { 1 } { 16 } = 2$
C) $\log _ { 4 } 16 = 2$
D) $\log _ { 4 } \frac { 1 } { 16 } = - 2$
E) $\log _ { 4 } 16 = - 2$

Accepted Answer

The exponential equation can be written as:
$$4^{-2}=\frac{1}{16}$$
Now, taking the logarithm of both sides with base 4:
$$\log_4(4^{-2})=\log_4(\frac{1}{16})$$
$$-2=\log_4(4^{-2})=\log_4(\frac{1}{16})$$
Hence, the correct choice is D.

Question 3

Evaluate the logarithm $\log _ { 1 / 3 } 1.877$ using the change of base formula. Round to 3 decimal places.
A) -1.745
B) -0.692
C) 0.630
D) 0.273
E) -0.573

Accepted Answer

E

Question 4

Evaluate the logarithm using the change-of-base formula. Round your result to three decimal places. $\log _ { 4 } 9$
A) $ 1.585 $
B) $ 5.585 $
C) $ 3.585 $
D) $ 4.585 $
E) $ 2.585 $

Accepted Answer

The change-of-base formula allows us to write $\log _ { 4 } 9$ as $\frac{\log 9}{\log 4}$ or $\frac{\ln 9}{\ln 4}$, where $\log$ is the logarithm base 10 and $\ln$ is the natural logarithm (base $e$). Using a calculator, this evaluates to approximately 1.585.

Question 5

Use the One-to-One Property to solve the equation for x. 
$\log _ { 2 } ( 8 x + 2 ) = \log _ { 2 } 14$ 
A) 8

B) $\frac { 1 } { 6 }$

C) $\frac { 3 } { 2 }$

D) $\frac { 1 } { 14 }$

E) $\frac { 1 } { 8 }$

Accepted Answer

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

\begin{align*}
\log_2(8x+2)&=\log_2(14)\
8x+2&=14\
8x&=12\
x&=\frac{3}{2}
\end{align*}

Therefore, the answer is (C).

Question 6

Solve the exponential equation algebraically. Approximate the result to three deci- mal places. $e ^ { x } - 8 = 12$ 
A) $\ln 20 \approx 2.485$
B) $\ln 20 \approx 2.996$
C) $\ln 20 \approx - 2.485$
D) $\ln 20 \approx 2.079$
E) $\ln 20 \approx - 2.996$

Accepted Answer

The answer of Solve the exponential equation algebraically. Approximate the...

Question 7

Use a graphing utility to construct a table of values for the function. Round your answer to two decimal places. 
$f ( x ) = 6 ^ { - x }$ 
A)
 $\begin{array}{|c|c|c|c|c|c|}
\hline x & -2 & -1 & 0 & 1 & 2 \
\hline f(x) & 36 & 6 & 1 & 0.17 & -0.03 \
\hline
\end{array}$

B)
$\begin{array}{|c|c|c|c|c|c|}
\hline x & -2 & -1 & 0 & 1 & 2 \
\hline f(x) & -36 & 6 & 1 & 0.17 & 0.03 \
\hline
\end{array}$

C)
$\begin{array}{|c|c|c|c|c|c|}
\hline x & -2 & -1 & 0 & 1 & 2 \
\hline f(x) & 36 & -6 & 1 & 0.17 & 0.03 \
\hline
\end{array}$

D)
$\begin{array}{|c|c|c|c|c|c|}
\hline x & -2 & -1 & 0 & 1 & 2 \
\hline f(x) & 36 & 6 & 1 & -0.17 & 0.03 \
\hline
\end{array}$

E)
$\begin{array}{|c|c|c|c|c|c|}
\hline x & -2 & -1 & 0 & 1 & 2 \
\hline f(x) & 36 & 6 & 1 & 0.17 & 0.03 \
\hline
\end{array}$

Accepted Answer

After using a graphing utility to plot the function, we can see that the values in choice E match the shape of the graph. Furthermore, the function is clearly positive for all values of x in the domain, so choices A, B, and C can be eliminated. Additionally, the function is decreasing as x increases, so choice D can also be eliminated. Therefore, the best choice is E.

Question 8

Find the exact value of the logarithmic expression without using a calculator. 
$\log _ { 9 } \sqrt [ 3 ] { 9 }$ 
A) 3
B) $\frac { 1 } { 3 }$
C) 9
D) 27
E) None of these

Accepted Answer

We can simplify the expression first:

$\log _ { 9 } \sqrt [ 3 ] { 9 } = \log _ { 9 } 9^{\frac13} = \frac13$

Therefore, the answer is (B).

Question 9

Evaluate the function at the indicated value of $x = a ^ { 2 }$ 
$g ( x ) = \log _ { a } x$ 
A) $- 2$
B) $\frac { 1 } { 2 }$
C) $- \frac { 1 } { 2 }$
D) 2
E) 0

Accepted Answer

If we substitute $a ^ { 2 }$ for x, we get $g(a^2)=\log_a(a^2)=2$. Therefore, the answer is choice D, 2.

Question 10

Evaluate the function at the indicated value of $X$ . Round your result to three decimal places. 
$\begin{array}{ll}\text { Function } & \text { Value } \ f(x)=0.5^{x} & x=1.7\end{array} $
A) $- 0.308$
B) $1.7$
C) $0.308$
D) $0.5$
E) $- 1.7$

Accepted Answer

Plugging in 1.7 for x, we get $0.5^{1.7} \approx 0.308$ rounding to 3 decimal places. Therefore, the answer is choice C.

Question 11

Rewrite the logarithm as a ratio of common logarithms. 
$\log _ { 5 } 19$ 
A) $\frac { \log 19 } { \log 5 }$

B) $\log \frac { 19 } { 5 }$

C) $\frac { \log 5 } { \log 19 }$

D) $\log \frac { 5 } { 19 }$

E) None of these

Accepted Answer

The base of the logarithm is 5, which is not a common base. We can use the change of base formula to express the logarithm as a ratio of common logarithms: $\log_5 19 = \frac{\log 19}{\log 5}$. Therefore, the answer is $\boxed{	ext{(A)}}$.

Question 12

Select the graph of the exponential function. 
$s ( t ) = 2 e ^ { 0.13 t }$ 
A)

B)

C)

D)

E)

Accepted Answer

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

Question 13

Determine the principal that must be invested at rate r, compounded monthly, so that $\$ 1,500,000$ will be available for retirement in t years. $r = 5 \% , t = 10$ (Round the answer upto 2 decimal places.) 
A)$\$ 1,110,741.56$
B) $\$ 1,010,741.56$
C) $\$ 710,741.56$
D) $\$ 810,741.56$
E) $\$ 910,741.56$

Accepted Answer

The answer of Determine the principal that must be invested...

Question 14

Write the exponential equation in logarithmic form. $e ^ { 4 x } = 5$ 
A) $\ln 5 = - 4 x$
B)$\ln 5 = - \frac { 1 } { 4 } x$
C)$\ln 5 = \frac { 1 } { 4 } x$
D) $\ln 5 = 4 x$
E) $\ln 4 = 5 x$

Accepted Answer

The exponential equation $e^{4x} = 5$ can be rewritten in logarithmic form as $\ln 5 = 4x$, using the definition of a logarithm.

Question 15

Determine whether the given x-value is a solution (or an approximate solution) of the equation. 
$\begin{array} { l } 
4 ^ { 5 x - 5 } = 1024 \
x = 2
\end{array}$

Accepted Answer

When we substitute x = 2 into the equation, we get:

4^(5(2)-5) = 4^5 = 1024

Since the left-hand side equals the right-hand side, x = 2 is a solution of the equation. Therefore, the answer is A, true.

Question 16

Condense the expression to the logarithm of a single quantity. $$\log x - 2 \log y + 3 \log z$$ 
A)$\log \frac { z ^ { 3 } } { x y ^ { 2 } }$
B) $\log \frac { x } { y ^ { 2 } z ^ { 3 } }$
C) $\log \frac { x y ^ { 2 } } { z ^ { 3 } }$
D) $\log \frac { y ^ { 2 } } { x z ^ { 3 } }$
E) $\log \frac { x z ^ { 3 } } { y ^ { 2 } }$

Accepted Answer

Using the properties of logarithms, the expression can be rewritten as $\log x - \log y^2 + \log z^3$, which simplifies to $\log \frac{xz^3}{y^2}$.

Question 17

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) $8.07$
B) $0.875$
C) $1.398$
D) $7.5$
E) $0.417$

Accepted Answer

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

Question 18

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. 
$$6 \log _ { 3 } ( 0.5 x ) = 5$$ 
A) $2 \left( 3 ^ { 5 / 6 } \right) \approx 3.996$
B) $2 \left( 3 ^ { 5 / 6 } \right) \approx 6.996$
C) $2 \left( 3 ^ { 5 / 6 } \right) \approx 4.996$
D) $2 \left( 3 ^ { 5 / 6 } \right) \approx 5.996$
E) $2 \left( 3 ^ { 5 / 6 } \right) \approx 2.996$

Accepted Answer

The answer of Solve the logarithmic equation algebraically. Approximate the...

Question 19

Write the logarithmic equation in exponential form. $$\log _ { 8 } 64 = 2$$ 
A) $64 ^ { 8 } = 2$
B) $8 ^ { 2 } = 16$
C) $8 ^ { 2 } = 88$
D) $8 ^ { 2 } = 64$
E) $8 ^ { 64 } = 2$

Accepted Answer

The logarithmic equation $\log _ { 8 } 64 = 2$ means that 64 is the 8th power of some number, and that number is 2. In other words, $\log _ { 8 } 64 = 2$ can be rewritten as $8^2 = 64$, which is the exponential form of the equation. Choice D matches this form, while the other choices do not.

Question 20

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) $\sqrt { 3 } \approx 2$
D) $\sqrt { 3 } \approx 1.732$
E) $\sqrt { 3 } = 1.732$

Accepted Answer

Using the definition of logarithms, we have:
$$\log_{3} x = \frac{1}{2} \iff 3^{\frac{1}{2}}=\sqrt{3}=x$$
Approximating $\sqrt{3}$ to three decimal places gives $1.732$, which matches option D).

Quiz 3: Exponential and Logarithmic Functions

Solve the exponential equation algebraically. Approximate the result to three decimal places. $e ^ { 2 x } - 5 e ^ { x } - 6 = 0$

Write the exponential equation in logarithmic form. $4 ^ { - 2 } = \frac { 1 } { 16 }$

Evaluate the logarithm $\log _ { 1 / 3 } 1.877$ using the change of base formula. Round to 3 decimal places.

Evaluate the logarithm using the change-of-base formula. Round your result to three decimal places. $\log _ { 4 } 9$

Use the One-to-One Property to solve the equation for x. $\log _ { 2 } ( 8 x + 2 ) = \log _ { 2 } 14$

Solve the exponential equation algebraically. Approximate the result to three deci- mal places. $e ^ { x } - 8 = 12$

Use a graphing utility to construct a table of values for the function. Round your answer to two decimal places. $f ( x ) = 6 ^ { - x }$

Find the exact value of the logarithmic expression without using a calculator. $\log _ { 9 } \sqrt [ 3 ] { 9 }$

Evaluate the function at the indicated value of $x = a ^ { 2 }$ $g ( x ) = \log _ { a } x$

Evaluate the function at the indicated value of $X$ . Round your result to three decimal places. $\begin{array}{ll}\text { Function } & \text { Value } \\ f(x) =0.5^{x} & x=1.7\end{array}$

Rewrite the logarithm as a ratio of common logarithms. $\log _ { 5 } 19$

Select the graph of the exponential function. $s ( t ) = 2 e ^ { 0.13 t }$

Determine the principal that must be invested at rate r, compounded monthly, so that $\$ 1,500,000$ will be available for retirement in t years. $r = 5 \% , t = 10$ (Round the answer upto 2 decimal places.)

Write the exponential equation in logarithmic form. $e ^ { 4 x } = 5$

Determine whether the given x-value is a solution (or an approximate solution) of the equation. $\begin{array} { l } 4 ^ { 5 x - 5 } = 1024 \\x = 2\end{array}$

Condense the expression to the logarithm of a single quantity. $\log x - 2 \log y + 3 \log z$

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

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $6 \log _ { 3 } ( 0.5 x ) = 5$

Write the logarithmic equation in exponential form. $\log _ { 8 } 64 = 2$

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

Quiz 3: Exponential and Logarithmic Functions

Solve the exponential equation algebraically. Approximate the result to three decimal places. e2x−5ex−6=0e ^ { 2 x } - 5 e ^ { x } - 6 = 0e2x−5ex−6=0

Write the exponential equation in logarithmic form. 4−2=1164 ^ { - 2 } = \frac { 1 } { 16 }4−2=161​

Evaluate the logarithm log⁡1/31.877\log _ { 1 / 3 } 1.877log1/3​1.877 using the change of base formula. Round to 3 decimal places.

Evaluate the logarithm using the change-of-base formula. Round your result to three decimal places. log⁡49\log _ { 4 } 9log4​9

Use the One-to-One Property to solve the equation for x. log⁡2(8x+2) =log⁡214\log _ { 2 } ( 8 x + 2 ) = \log _ { 2 } 14log2​(8x+2) =log2​14

Solve the exponential equation algebraically. Approximate the result to three deci- mal places. ex−8=12e ^ { x } - 8 = 12ex−8=12

Use a graphing utility to construct a table of values for the function. Round your answer to two decimal places. f(x) =6−xf ( x ) = 6 ^ { - x }f(x) =6−x

Find the exact value of the logarithmic expression without using a calculator. log⁡993\log _ { 9 } \sqrt [ 3 ] { 9 }log9​39​

Evaluate the function at the indicated value of x=a2x = a ^ { 2 }x=a2 g(x) =log⁡axg ( x ) = \log _ { a } xg(x) =loga​x

Evaluate the function at the indicated value of XXX . Round your result to three decimal places. Function Value f(x) =0.5xx=1.7\begin{array}{ll}\text { Function } & \text { Value } \\ f(x) =0.5^{x} & x=1.7\end{array} Function f(x) =0.5x​ Value x=1.7​

Rewrite the logarithm as a ratio of common logarithms. log⁡519\log _ { 5 } 19log5​19

Select the graph of the exponential function. s(t) =2e0.13ts ( t ) = 2 e ^ { 0.13 t }s(t) =2e0.13t

Determine the principal that must be invested at rate r, compounded monthly, so that $1,500,000\$ 1,500,000$1,500,000 will be available for retirement in t years. r=5%,t=10r = 5 \% , t = 10r=5%,t=10 (Round the answer upto 2 decimal places.)

Write the exponential equation in logarithmic form. e4x=5e ^ { 4 x } = 5e4x=5

Determine whether the given x-value is a solution (or an approximate solution) of the equation. 45x−5=1024x=2\begin{array} { l } 4 ^ { 5 x - 5 } = 1024 \\x = 2\end{array}45x−5=1024x=2​

Condense the expression to the logarithm of a single quantity. log⁡x−2log⁡y+3log⁡z\log x - 2 \log y + 3 \log zlogx−2logy+3logz

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​

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. 6log⁡3(0.5x) =56 \log _ { 3 } ( 0.5 x ) = 56log3​(0.5x) =5

Write the logarithmic equation in exponential form. log⁡864=2\log _ { 8 } 64 = 2log8​64=2

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