Exam 5: Inverse, Exponential, and Logarithmic Functions

solve each equation. Give the solution set (a) with an exact value and then (b) with an approximation to the nearest thousandth. - $\ln (\log 3 x)=-1$

Consider the equation $\log _{2} x+\log _{2}(x-4)=5$ . (a) Solve the equation analytically. If there is an extraneous value, what is it? (b) To support the solution in part (a), we may graph $y_{1}=\log _{2} x+\log _{2}(x-4)-5$ and find the $x$ -intercept. Write an expression for $y_{1}$ using the change-of-base rule with base 10 , and graph the function to support the solution from part (a). (c) Use the graph to solve the inequality $\log _{2} x+\log _{2}(x-4)<5$ .

The concentration of pollutants in a stream is given by $y=.06 e^{-3 x}$ , where $y$ is the amount of pollutant in grams per liter and $x$ is the distance, in kilometers, downstream from the source of the pollution. Match each question with one of the solutions A, B, C, or D. -How far downstream is the pollutant level equal to $.02 \mathrm{gm} / \mathrm{L}$ ?

An unstable radioactive isotope decays according to the equation $y=2.53(.97)^{t}$ where $y$ is the number of grams remaining and $t$ is the time measured in minutes. Match each question with one of the solutions A, B, C, or D. -How long will it take for the material to decay to half its initial amount?

Consider the equation $\log _{6} x+\log _{6}(x-5)=1$ . (a) Solve the equation analytically. If there is an extraneous value, what is it? (b) To support the solution in part (a), we may graph $y_{1}=\log _{6} x+\log _{6}(x-5)-1$ and find the $x$ -intercept. Write an expression for $y_{1}$ using the change-of-base rule with base 10, and graph the function to support the solution from part (a). (c) Use the graph to solve the inequality $\log _{6} x+\log _{6}(x-5)<1$ .

solve each equation. Give the solution set (a) with an exact value and then (b) with an approximation to the nearest thousandth. - $\log (\ln x)=0$

Use the power, quotient, and product properties of logarithms to write $\log \frac{p^{3} q^{2}}{\sqrt[4]{r}}$ as an equivalent expression.

solve each equation. Give the solution set (a) with an exact value and then (b) with an approximation to the nearest thousandth. - $5^{2 x+1}=3^{4 x-1}$

Solve $A=P e^{r t}$ for $r$ .

Use the power, quotient, and product properties of logarithms to write $\log \frac{a}{\sqrt{b c^{3}}}$ as an equivalent expression.

A sample of radioactive material has a half-life of about 1250 years. An initial sample weighs 12 grams. (a) Find a formula for the decay function for this material. (b) Find the amount left after 5000 years. (c) Find the time for the initial amount to decay to 3 grams.

Use a calculator to find an approximation of each logarithm to the nearest thousandth. (a) $\log 17.6$ (b) $\log _{9} 750$ (c) $\ln 901$

Suppose that $\$ 25,000$ is invested at $5.7 \%$ for 12 years. Find the total amount present at the end of thime period if the interest is compounded (a) quarterly and (b) continuously.

Match each equation with its graph. - $y=e^{x}$

Match each equation with its graph. - $y=\ln x$

Match each equation with its graph. - $y=e^{x}$

One of your friends is taking another mathematics course and tells you, "I have no idea what an expression like $\log _{7} 12$ really means." Write an explanation of what it means, and tell how you can find an approximation for it with a calculator.

Solve the equation $9^{x+3}=\left(\frac{1}{3}\right)^{x-7}$ analytically.

Match each equation with its graph. - $y=\ln x$

Match each equation with its graph - $y=e^{x}$