Exam 9: Logarithmic and Exponential Functions

Graph. State the domain, range, and vertical asymptote of the function - $f(x)=\log _{5}(x+1)-2$

Solve the equation. - $\log (3 x-1)-\log 5 x=3$

Evaluate the given function - $f(x)=3^{x+2}+2, f(3)$

Expand. A ssume that all variables represent positive real numbers. - $\log _{b}\left(\frac{x^{2}}{z^{9}}\right)$

Use the horizontal-line test to determine whether the function is one to one. -

Graph the given function f(x). Label the vertical asymptote. State the domain and range of the function. - $f(x)=\log x$

Determine the equation of the horizontal asymptote for the graph of this function, and state the domain and range of thisfunction. - $f(x)=2^{x+3}$

Use the horizontal-line test to determine whether the function is one to one. -

Find the missing number. Round to the nearest hundredth if necessary. - $6^ ?=11$

Graph the piecewise function - $f(x)=\left\{\begin{array}{lr}\log (x-3)+1 & \text { if } x>3 \\ (x-3)^{2} & \text { if } x \leq 3\end{array}\right.$

The area betw een the graph of $f(x)=e^{x}$ and the $x$ -axis on an interval $[a, b]$ is equal to $e^{b}-e^{a}$ . Sketch the graph of $f(x)=e^{x}$ , and shade the area between the graph and the $x$ -axis. Find the shaded area, rounded to the nearest hundredth. - $[-2,1]$

Use the formula $\frac{f(b)-f(a)}{b-a}$ to find the average rate of change of the function f(x) on the interval [a, b]. - $f(x)=7^{x},[0,2]$

Simplify. - $\log _{8} 8^{2}$

Another way to approximate the value of $e$ is through the sum $1+\frac{1}{1 !}+\frac{1}{2 !}+\frac{1}{3 !}+\ldots$ . The notation $n !$ represents $n$ factorial, which is the product of all the integers from 1 through $n$ . The more terms added to the sum, the closer the sum gets to e. In calculus, we say that the limit of this sum is e. Find the sum of the first $k$ terms, rounded to the nearest hundred-thousandth, and determine how close the result is to e rounded to the nearest hundred-thousandth (2.71828). - $\mathrm{k}=9$ terms

Solve the problem. -A college loan of $\$ 25,000$ is made at $4 \%$ interest, compounded annually. After tyears, the amount due, $\mathrm{A}$ , is given by the function $\mathrm{A}(\mathrm{t})=25,000(1.04)^{\mathrm{t}}$ After what amount of time will the amount reach $\$ 37,000$ ? Round your answer to the nearest tenth.

Determine whether the graph is the graph of a function. -

Solve the problem. -Approximately one fourth of all glass bottles distributed will be recycled each year. A beverage company distributes 310,000 bottles. The number still in use after $t$ years is given by the function $\mathrm{N}(\mathrm{t})=310,000\left(\frac{1}{4}\right)^{\mathrm{t}}$ After how many years will 30,000 bottles still be in use? Round your answer to the nearest tenth.

Solve. -The function $\mathrm{A}=\mathrm{A}_{0} \mathrm{e}^{-0.0077 \mathrm{x}}$ models the amount in pounds of a particular radioactive material stored in a concrete vault, where $\mathrm{x}$ is the number of years since the material was put into the vault. If 900 pounds of the material are initially put into the vault, how many pounds will be left after 80 years?

Solve. - $2^{\mathrm{X}-1}=18$ Round to the nearest thousandth.

Evaluate the given function - $f(x)=\left(\frac{1}{2}\right)^{x+2}+5, f(2)$