Question 1

Find the following using a calculator. Round to four decimal places.
-$\log 93,500$

Accepted Answer

The answer of Find the following using a calculator. Round...

Question 2

Express as a difference of logarithms.
-$\log _ { 4 } \frac { 9 } { 11 }$

Accepted Answer

The logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator, hence $\log _ { 4 } \frac { 9 } { 11 } = \log _ { 4 } 9 - \log _ { 4 } 11$.

Question 3

Graph the piecewise function.
-..

Accepted Answer

The answer of Graph the piecewise function.
-..
 ...

Question 4

Solve.
-Given $\log _ { \mathrm { b } } 2 = 0.5298$ and $\log _ { \mathrm { b } } 7 = 1.4873$, evaluate $\log _ { \mathrm { b } } 2 \mathrm {~b}$.

Accepted Answer

The answer of Solve.
-Given \(\log _ { \mathrm { b...

Question 5

Evaluate to four decimal places using a calculator.
-$\mathrm { e } ^ { 1.233 }$

Accepted Answer

The answer of Evaluate to four decimal places using a...

Question 6

Solve the problem.
-Suppose the amount of a radioactive element remaining in a sample of 100 milligrams after $x$ years can be described by $\mathrm { A } ( \mathrm { x } ) = 100 \mathrm { e } ^ { - 0.01569 \mathrm { x } }$. How much is remaining after 58 years? Round the answer to the nearest hundredth of a milligram.

Accepted Answer

To find the amount remaining after 58 years, substitute $x = 58$ into the given formula: $A(58) = 100e^{-0.01569 \times 58}$. Calculating this gives approximately $40.25$ milligrams.

Question 7

Convert to a logarithmic equation.
-$$4 ^ { 2 } = 16$$

Accepted Answer

The logarithmic form of $a^b = c$ is $b = \log_a c$. Therefore, converting $4^2 = 16$ to logarithmic form gives $2 = \log_4 16$.

Question 8

Graph the function.
-$$f ( x ) = \left( \frac { 1 } { 5 } \right) ^ { x }$$

Accepted Answer

The answer of Graph the function.
-\[f ( x ) =...

Question 9

Convert to a logarithmic equation.
-$10 ^ { 0.6021 } = 4$

Accepted Answer

The logarithmic form of $10 ^ { 0.6021 } = 4$ is $0.6021 = \log _ { 10 } 4$, which matches choice D. This is because the base of the logarithm corresponds to the base of the exponent, and the exponent itself becomes the result of the logarithm.

Question 10

Graph the function.
-$f(x)=4(x-3)+3$

Accepted Answer

The answer of Graph the function.
-$f(x)=4(x-3)+3$

...

Question 11

Graph the function.
-$$y = \frac { 1 } { 6 } e ^ { x }$$

Accepted Answer

The answer of Graph the function.
-\[y = \frac { 1...

Question 12

Graph the function. Describe its position relative to the graph of the indicated basic function.
-

Accepted Answer

The answer of Graph the function. Describe its position relative...

Question 13

The graph of a one-to-one function f is given. Sketch the graph of the inverse function $$\mathrm { f } ^ { - 1 }$$ , on the same set of axes. Use a
dashed line for the inverse.
-

Accepted Answer

The answer of The graph of a one-to-one function f...

Question 14

Find the domain and range of the inverse of the given function.
-$$f ( x ) = x ^ { 2 } - 8 ; x \geq 0$$

Accepted Answer

The domain of the inverse is the range of the original function, which is $[ - 8 , \infty )$. The range of the inverse is the domain of the original function, which is $[ 0 , \infty )$.

Question 15

Find the domain and the vertical asymptote of the function.
-$$f ( x ) = - 2 - 4 \log ( x + 1 )$$

Accepted Answer

The argument of the logarithm, $x + 1$, must be greater than 0, so $x > -1$, which gives the domain $(-1, \infty)$. The vertical asymptote occurs where the argument of the logarithm equals 0, which is at $x = -1$.

Question 16

Graph the piecewise function.
-$$f ( x ) = \left\{ \begin{array} { l l } 
e ^ { - x } - 8 , & \text { for } x < - 2 \
x - 2 , & \text { for } - 2 \leq x < 1 \
x ^ { 3 } , & \text { for } x \geq 1
\end{array} \right.$$

Accepted Answer

The answer of Graph the piecewise function.
-\[f ( x )...

Question 17

Solve the problem.
-An initial investment of $14,000 is appreciated for 4 years in an account that earns 13% interest, compounded semiannually. Find the amount of money in the account at the end of the period.

Accepted Answer

The formula for compound interest is A = P(1 + r/n)^(nt), where P is the principal amount ($14,000), r is the annual interest rate (0.13), n is the number of times interest is compounded per year (2 for semiannually), and t is the number of years (4). Plugging in these values gives A = 14000(1 + 0.13/2)^(2*4) = $23,169.94.

Question 18

Solve.
-Find the hydrogen ion concentration of a solution whose $\mathrm { pH }$ is $6.8$. Use the formula $\mathrm { pH } = - \log \left[ \mathrm { H } ^ { + } \right]$.

Accepted Answer

The formula to find the hydrogen ion concentration from pH is $[H^+] = 10^{-\text{pH}}$. Substituting $6.8$ for pH, we get $[H^+] = 10^{-6.8} = 1.58 \times 10^{-7}$.

Question 19

Solve the problem.
-In September 1998 the population of the country of West Goma in millions was modeled by $\mathrm { f } ( \mathrm { x } ) = 16.6 \mathrm { e } ^ { 0.0012 \mathrm { x } }$. At the same time the population of East Goma in millions was modeled by $\mathrm { g } ( \mathrm { x } ) = 13.4 \mathrm { e } ^ { 0.0133 \mathrm { x } }$. In both formulas $\mathrm { x }$ is the year, where $\mathrm { x } = 0$ corresponds to September $1998 .$ Assuming these trends continue, estimate the year when the population of West Goma will equal the population of East Goma.

Accepted Answer

To find when the populations are equal, set $16.6 e^{0.0012x} = 13.4 e^{0.0133x}$. Solving for $x$ gives approximately 18 years after 1998, which is 2016.

Question 20

Find the domain and the vertical asymptote of the function.
-$$f ( x ) = \log ( x - 10 )$$

Accepted Answer

The logarithmic function $\log(x - 10)$ is defined for $x - 10 > 0$, which gives $x > 10$, so the domain is $(10, \infty)$. The vertical asymptote occurs where the argument of the logarithm is zero, which is at $x = 10$.

Quiz 5: Exponential and Logarithmic Functions

Find the following using a calculator. Round to four decimal places. - $\log 93,500$

Express as a difference of logarithms. - $\log _ { 4 } \frac { 9 } { 11 }$

Graph the piecewise function. -..

Solve. -Given $\log _ { \mathrm { b } } 2 = 0.5298$ and $\log _ { \mathrm { b } } 7 = 1.4873$ , evaluate $\log _ { \mathrm { b } } 2 \mathrm {~b}$ .

Evaluate to four decimal places using a calculator. - $\mathrm { e } ^ { 1.233 }$

Convert to a logarithmic equation. - $4 ^ { 2 } = 16$

Graph the function. - $f ( x ) = \left( \frac { 1 } { 5 } \right) ^ { x }$ $Graph the function. - f ( x ) = \left( \frac { 1 } { 5 } \right) ^ { x } A) B) C) D)$

Convert to a logarithmic equation. - $10 ^ { 0.6021 } = 4$

Graph the function. - $f(x) =4(x-3) +3$

Graph the function. - $y = \frac { 1 } { 6 } e ^ { x }$ $Graph the function. - y = \frac { 1 } { 6 } e ^ { x } A) B) C) D)$

Graph the function. Describe its position relative to the graph of the indicated basic function. -

Find the domain and range of the inverse of the given function. - $f ( x ) = x ^ { 2 } - 8 ; x \geq 0$

Find the domain and the vertical asymptote of the function. - $f ( x ) = - 2 - 4 \log ( x + 1 )$

Solve the problem. -An initial investment of $14,000 is appreciated for 4 years in an account that earns 13% interest, compounded semiannually. Find the amount of money in the account at the end of the period.

Solve. -Find the hydrogen ion concentration of a solution whose $\mathrm { pH }$ is $6.8$ . Use the formula $\mathrm { pH } = - \log \left[ \mathrm { H } ^ { + } \right]$ .

Find the domain and the vertical asymptote of the function. - $f ( x ) = \log ( x - 10 )$

Quiz 5: Exponential and Logarithmic Functions

Find the following using a calculator. Round to four decimal places. - log⁡93,500\log 93,500log93,500

Express as a difference of logarithms. - log⁡4911\log _ { 4 } \frac { 9 } { 11 }log4​119​

Graph the piecewise function. -..

Solve. -Given log⁡b2=0.5298\log _ { \mathrm { b } } 2 = 0.5298logb​2=0.5298 and log⁡b7=1.4873\log _ { \mathrm { b } } 7 = 1.4873logb​7=1.4873 , evaluate log⁡b2 b\log _ { \mathrm { b } } 2 \mathrm {~b}logb​2 b .

Evaluate to four decimal places using a calculator. - e1.233\mathrm { e } ^ { 1.233 }e1.233

Convert to a logarithmic equation. - 42=164 ^ { 2 } = 1642=16

Graph the function. - f(x) =(15) xf ( x ) = \left( \frac { 1 } { 5 } \right) ^ { x }f(x) =(51​) x

Convert to a logarithmic equation. - 100.6021=410 ^ { 0.6021 } = 4100.6021=4

Graph the function. - f(x) =4(x−3) +3f(x) =4(x-3) +3f(x) =4(x−3) +3

Graph the function. - y=16exy = \frac { 1 } { 6 } e ^ { x }y=61​ex