Exam 10: Inverse, Exponential, and Logarithmic Functions

The total number $N$ of bacteria cells in a petri dish when exposed to a new drug can be approximated by the function $\mathrm{N}(\mathrm{x})=75 \mathrm{e}^{-0.02532 \mathrm{x}}$ , where $\mathrm{N}$ is measured in thousands; and $\mathrm{x}=0$ corresponds to the initial number of cells, $\mathrm{x}=1$ corresponds to the number of cells after one hour, and so on. The function is graphed on a graphing calculator-generated screen. Interpret the meanings of $x$ and $y$ in the display at the bottom of the screen.

If the following defines a one-to-one function, find its inverse. If not, write "Not one-to-one." - $f(x)=\frac{8}{x-9}$

Solve the equation. - $\log _{x} 625=4$

A student argues that the best choice for the logarithmic base to be used in solving $7 \mathrm{X}=50$ is 7 . What is wrong with this thinking?

Jorge invested $\$ 7500$ at $8 \%$ compounded semiannually. In how many years will Jorge's investment have tripled? Round your answer to the nearest tenth of a year.

If the following defines a one-to-one function, find its inverse. If not, write "Not one-to-one." - $f(x)=4 x-6$

Solve the equation. - $4^{(1+2 \mathrm{x})}=1024$

If the following defines a one-to-one function, find its inverse. If not, write "Not one-to-one." - $f(x)=x^{3}-5$

Barry Newman's savings account has a balance of $\$ 1964$ . After 10 years, what will the amount of interest be at $8 \%$ compounded annually?

Write in exponential form. - $\log _{-2}-8=\mathrm{x}$

A certain noise has an intensity $I$ of $1.91 \times 10^{-5}$ . Given that decibel level $D$ is related to intensity by $\mathrm{D}=10 \log \left(\frac{\mathrm{I}}{\mathrm{I}_{\mathrm{O}}}\right)$ , where $\mathrm{I}_{\mathrm{O}}$ is $10^{-12}$ , determine the decibel level of the noise. Round your answer to the nearest decibel.

The population of a particular city is increasing at a rate proportional to its size. It follows the function $\mathrm{P}(\mathrm{t})=1+\mathrm{ke} 0.08 \mathrm{t}$ where $\mathrm{k}$ is a constant and $\mathrm{t}$ is the time in years. If the current population is 12,000 , in how many years is the population expected to be 30,000 ? (Round to the nearest year.)