Exam 3: Exponential, Logistic, and Logarithmic Functions

Solve the problem. -In the formula $\mathrm { A } = \mathrm { Ie } \mathrm { kt } , \mathrm { A }$ is the amount of radioactive material remaining from an initial amount I at a given time $t$ and $k$ is a negative constant determined by the nature of the material. A certain radioactive isotope decays at a rate of $0.175 \%$ annually. Determine the half-life of this isotope, to the nearest year.

Simplify the expression. - $10 ^ { \log ( 1 / 7 ) }$

Solve the problem. -Find the periodic payment of a loan with present value $1500 and an annual interest rate 8.9% for a term of 3 years, with payments made and interest charged 12 times per year.

Use the change of base rule to find the logarithm to four decimal places. - $\log _ { \sqrt { 7 } } 285.5$

Find the following using a calculator. Round to four decimal places. - $\ln 153$

Determine a formula for the exponential function. -The indicated point is $\left( 0 , \mathrm { e } ^ { - 1 } \right)$ . Which of the following is the correct equation of the function?

Use a calculator to find an approximate solution to the equation. - $3 x = 25$

Describe how to transform the graph of the basic function g(x) into the graph of the given function f(x). - $f ( x ) = - \log ( x + 6 ) ; \quad g ( x ) = \log x$

Solve the equation by changing it to exponential form. - $\log _ { 16 } x = \frac { 1 } { 2 }$

Write the expression using only the indicated logarithms. - $\log _ { 3 } ( x + y )$ using natural logarithms

Determine the doubling time of the investment. -7% APR compounded semiannually

Find the following using a calculator. Round to four decimal places. - $\log 0.85$

Solve the problem. -In the formula $\mathrm { A } =$ Iekt, $\mathrm { A }$ is the amount of radioactive material remaining from an initial amount I at a given time $t$ and $\mathrm { k }$ is a negative constant determined by the nature of the material. An artifact is discovered at a certain site. If it has $54 \%$ of the carbon-14 it originally contained, what is the approximate age of the artifact? (carbon-14 decays at the rate of $0.0125 \%$ annually.) (Round to the nearest year.)

Solve the problem. -Assume the cost of a gallon of milk is $2.50. With continuous compounding, find the time it would take the cost to be 2 times as much (to the nearest tenth of a year), at an annual inflation rate of 6%.

Graph the function. Describe its position relative to the graph of the indicated basic function. - $f ( x ) = \ln ( x + 5 ) \text {; relative to } f ( x ) = \ln x$

Evaluate the logarithm. - $\log _ { 9 } \left( \frac { 1 } { 81 } \right)$

Solve the problem. -The number of books in a small library increases according to the function $\mathrm { B } = 6700 \mathrm { e } 0.02 \mathrm { t }$ , where $t$ is measured in years. How many books will the library have after 3 years?

Assuming all variables are positive, use properties of logarithms to write the expression as a sum or difference of logarithms or multiples of logarithms. -log 6 12x

Solve the problem. -Find the interest rate necessary for a present value of $18,401 to grow to a future value of $21,559.70 if interest is compounded quarterly for two years.

Solve the problem. -There are currently 57 million cars in a certain country, increasing by $7.5 \%$ annually. How many years will it take for this country to have 85 million cars? Round to the nearest year.