Exam 5: Inverse, Exponential, and Logarithmic Functions

Use the power, quotient, and product properties of logarithms to write $\ln \frac{\sqrt[3]{x^{2} y}}{z^{4}}$ as an equivalent expression.

Consider the function $f(x)=-4^{x+1}+3$ . (a) Graph it in the standard viewing window of your calculator. (b) Give the domain and range of $f$ . (c) Does the graph have an asymptote? If so, is it vertical or horizontal, and what is its equation? (d) Find the $x$ - and $y$ -intercepts analytically and use the graph from part (a) to support your answers graphically. (e) Find $f^{-1}(x)$ .

Match each equation with its graph. - $y=\log _{1 / 3} x$

A water tank has been contaminated with salt and must be flushed with pure water. The concentration of salt in the tank is given by the equation $y=.1+3.02 e^{-.4 t}$ , where $y$ is the concentration of salt in milligrams per gallon and $t$ is the time, in hours, since flushing began. Match each question with one of the solutions $\mathrm{A}, \mathrm{B}, \mathrm{C}$ , or $\mathrm{D}$ . -What is the concentration of salt after 30 minutes?

Solve the equation $\left(\frac{1}{36}\right)^{2 x-3}=6^{3 x+1}$ analytically.

Use a calculator to find an approximation of each logarithm to the nearest thousandth. (a) $\ln 21.6$ (b) $\log _{3} 19.1$ (c) $\log 153$

Solve $T=T_{0}+C e^{-k t}$ for $t$ .

solve each equation. Give the solution set (a) with an exact value and then (b) with an approximation to the nearest thousandth. - $10^{1-x}=3^{2 x+1}$

Consider the function $f(x)=-2^{x-3}-1$ . (a) Graph it in the standard viewing window of your calculator. (b) Give the domain and range of $f$ . (c) Does the graph have an asymptote? If so, is it vertical or horizontal, and what is its equation? (d) Find the $x$ - and $y$ -intercepts analytically and use the graph from part (a) to support your answers graphically. (e) Find $f^{-1}(x)$ .

solve each equation. Give the solution set (a) with an exact value and then (b) with an approximation to the nearest thousandth. - $12^{2 x-2}=6^{2 x-1}$

A water tank has been contaminated with salt and must be flushed with pure water. The concentration of salt in the tank is given by the equation $y=.1+3.02 e^{-.4 t}$ , where $y$ is the concentration of salt in milligrams per gallon and $t$ is the time, in hours, since flushing began. Match each question with one of the solutions $\mathrm{A}, \mathrm{B}, \mathrm{C}$ , or $\mathrm{D}$ . -How long will it take for the concentration to reach half its initial value?

When a loaf of bread is removed from the freezer to thaw, the temperature of the bread increases. Graph each of the following functions on the interval $[0,50]$ . Use $[0,100]$ for the range of $A(t)$ . Use a graphing calculator to determine the function that best describes the temperature of the bread $A(t)$ (in degrees Fahrenheit) $t$ minutes after it is removed from the freezer, if the initial temperature of the bread was 30 degrees Fahrenheit. (a) $A(t)=-.2 t^{2}+4 t+30$ (b) $A(t)=70-40 e^{.01 t}$ (c) $A(t)=30 \ln (t+1)$ (d) $A(t)=70-40 e^{-.06 t}$

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. -What is the pollutant level .02 kilometer from the source?

One of your friends is taking another mathematics course and tells you, "I have no idea what an expression like $\log _{6} 27$ really means." Write an explanation of what it means, and tell how you can find an approximation for it with a calculator.

A sample of radioactive material has a half-life of about 1600 years. An initial sample weighs 18 grams. (a) Find a formula for the decay function for this material. (b) Find the amount left after 6400 years (c) Find the time for the initial amount to decay to 4.5 grams.

Match each equation with its graph. - $y=\left(\frac{1}{2}\right)^{x}$

Match each equation with its graph. - $y=\left(\frac{1}{3}\right)^{x}$

Suppose that $\$ 8100$ is invested at 3.8 % for 20 years. Find the total amount present at the end of this time period if the interest is compounded (a) quarterly and (b) continuously.

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 half the amount at the source?

solve each equation. Give the solution set (a) with an exact value and then (b) with an approximation to the nearest thousandth. - $4 e^{5 x+1}=8$