Exam 4: Exponential and Logarithmic Functions

Use the compound interest formulas A $A = P \left( 1 + \frac { r } { n } \right) ^ { n t } \text { and } A = P e ^ { r t } \text { to solve. }$ -Find the accumulated value of an investment of $12,000 at 8% compounded semiannually for 11 years.

Solve the problem. -The function $f ( x ) = 1 + 1.6 \ln ( x + 1 )$ models the average number of free-throws a basketball player can make consecutively during practice as a function of time, where $x$ is the number of consecutive days the basketball player has practiced for two hours. After how many days of practice can the basketball player make an average of 8 consecutive free throws?

Graph the function by making a table of coordinates. - $f(x)=\left(\frac{3}{2}\right)^{x}$

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. - $\ln 4 + \ln ( x - 1 ) = 0$

Graph the function. - $\text { Use the graph of } \log _ { 5 } x \text { to obtain the graph of } f ( x ) = 2 + \log _ { 5 } x \text {. }$

Solve the problem. -If Emery has $2000 to invest at 8% per year compounded monthly, how long will it be before he has $3300? If the compounding is continuous, how long will it be? (Round your answers to three decimal places.)

Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions. - $\log x + \log \left( x ^ { 2 } - 25 \right) - \log 8 - \log ( x - 5 )$

Graph the functions in the same rectangular coordinate system. - $f(x)=2^{X} \text { and } g(x)=\log _{2} x$

Write the equation in its equivalent logarithmic form. - $10 ^ { 3 } = y$

Approximate the number using a calculator. Round your answer to three decimal places. -The formula $S = A \left( \frac { ( 1 + r ) ^ { t + 1 } - 1 } { r } \right)$ models the value of a retirement account, where $A =$ the number of dollars added to the retirement account each year, $\mathrm { r } =$ the annual interest rate, and $\mathrm { S } =$ the value of the retirement account after t years. If the interest rate is $6 \%$ , how much will the account be worth after 15 years if $\$ 400$ is added each year? Round to the nearest whole number.

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. - $\ln x = 5$

Evaluate or simplify the expression without using a calculator. -The $\mathrm { pH }$ of a solution ranges from 0 to 14 . An acid has a $\mathrm { pH }$ less than 7 . Pure water is neutral and has a $\mathrm { pH }$ of 7. The $\mathrm { pH }$ of a solution is given by $\mathrm { pH } = - \log \mathrm { x }$ where $\mathrm { x }$ represents the concentration of the hydrogen ions in the solution in moles per liter. Find the $\mathrm { pH }$ if the hydrogen ion concentration is $1 \times 10 ^ { - 4 }$ .

Evaluate the expression without using a calculator. - $\log _ { 3 } \frac { 1 } { \sqrt { 3 } }$

Use the compound interest formulas A $A = P \left( 1 + \frac { r } { n } \right) ^ { n t } \text { and } A = P e ^ { r t } \text { to solve. }$ -Find the accumulated value of an investment of $6000 at 8% compounded continuously for 4 years.

Write the equation in its equivalent exponential form. - $\log _ { 3 } 9 = x$

Solve. -The function $\mathrm { A } = \mathrm { A } _ { 0 } \mathrm { e } ^ { - 0.0077 \mathrm { x } }$ models the amount in pounds of a particular radioactive material stored in a concrete vault, where $x$ is the number of years since the material was put into the vault. If 300 pounds of the material are initially put into the vault, how many pounds will be left after 30 years?

Approximate the number using a calculator. Round your answer to three decimal places. -The size of the bear population at a national park increases at the rate of $4.8 \%$ per year. If the size of the current population is 152 , find how many bears there should be in 4 years. Use the function $f ( x ) = 152 e ^ { 0.048 t }$ and round to the nearest whole number.

Graph the function - $\text { Use the graph of } f ( x ) = e ^ { x } \text { to obtain the graph of } g ( x ) = - e ^ { x } \text {. }$

Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions. - $3 \ln ( x - 6 ) - 10 \ln x$

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - $\log _ { C } x ^ { 4 }$