Exam 10: Inverse, Exponential, and Logarithmic Functions

Solve the problem. Round your answer to the nearest tenth, when appropriate. Use the formula $\mathrm{pH}=-\log \left[\mathrm{H}_{3} \mathrm{O}^{+}\right]$ , as needed. -Find the $\mathrm{pH}$ if $\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=1.7 \times 10-11$ .

Use a calculator and the change-of-base formula to find the logarithm to four decimal places. - $\log _{6} 0.899$

The annual depreciation rate $r(0<r<1)$ of a car purchased for $P$ dollars and worth A dollars after $t$ years can be modeled by the following formula: $\log (1-r)=\frac{1}{t} \log \frac{A}{P}$ Find the depreciation rate of a car that is purchased for $\$ 35,000$ and is sold 3 years later for $\$ 22,000$ . Express your answer as a percentage, and round the answer to the nearest whole percentage.

Use properties of logarithms to write each expression as a sum or difference of logarithms. Assume that variables represent positive real numbers. - $\log _{7} \frac{\sqrt[9]{5}}{n^{2} m}$

Graph the given logarithmic function. - $y=\log _{1 / 3} x$

Solve the equation. Give the exact solution or solutions. - $\log _{\mathrm{x}} 9=5$

Can you solve this equation without using logarithms? Why or why not? $5=3 x$

Solve the equation. Use natural logarithms. When appropriate, give solutions to three decimal places unless otherwise indicated. - $\ln \mathrm{e}^{\mathrm{X}}=11$

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

Solve the equation. - $4^{\mathrm{x}}=\frac{1}{64}$

Use a calculator and the change-of-base formula to find the logarithm to four decimal places. - $\log _{4} 8$

Solve the equation. - $2-x=\frac{1}{16}$

Solve the equation. Give the solution to three decimal places. - $25 \mathrm{x}-2=22$

Using the exponential key of a calculator to find an approximation to the nearest thousandth. - $2.568^{-3.8}$

Write in exponential form. - $\log _{1 / 5} 125=-3$

Express the given logarithm as a sum and/or difference of logarithms. Simplify, if possible. Assume that all variables represent positive real numbers. - $\log _{5} \frac{x^{2} y^{7}}{9}$

The number of bacteria growing in an incubation culture increases with time according to $B=9500(5)^{x}$ , where $x$ is time in days. Find the number of bacteria when $x=0$ and $x=2$ .

Coyotes are one of the few species of North American animals with an expanding range. The future population $P$ of coyotes in a region of Mississippi can be modeled by the equation $P(t)=57+18 \ln (14 t+1)$ , where $t$ is time in years. How long will it take for the population to reach 180 ? Round your answer to the nearest tenth, if necessary.

Solve the problem. Round your answer to the nearest tenth, when appropriate. Use the formula $\mathrm{pH}=-\log \left[\mathrm{H}_{3} \mathrm{O}^{+}\right]$ , as needed. -Find $\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]$ if the $\mathrm{pH}=12$ .

What will be the amount in an account with initial principal $\$ 6000$ if interest is compounded continuously at an annual rate of $5.25 \%$ for 7 years?