Exam 10: Inverse, Exponential, and Logarithmic Functions

Fill in the blanks with the correct responses. -The value of $\log _{5} 125$ is _______. This means that if we raise ______ to the power ______, the result is _______

Write in logarithmic form. - $4^{3 / 2}=8$

When solving an exponential equation using logarithms, when is it easier to use natural logarithms?

Find the amount of money in an account after 3 years if $\$ 2900$ is deposited at $6 \%$ annual interest compounded monthly.

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

The logarithmic function $f(x)=-100+80 \ln x$ models the number of visitors (in millions) to U.S. museums from 1940 to 1990, where $\mathrm{x}$ is the number of years since 1900. Use this function to estimate the number of visitors in the year 1985.

Solve the equation. Give the solution to three decimal places. - $3 x-2=11$

Solve the equation. Give the exact solution or solutions. - $\log (2 x-1)=\log 1-\log (x-1)$

Quail are game birds that fare poorly when their habitat is encroached upon. Wildlife biologists have discovered that the population $P$ of quail in a region is related to the percent of the region that has been paved with roads and parking lots, according to the formula $P=k \ln \left(\frac{30}{x+3}\right), 0 \leq x \leq 27$ Where $\mathrm{x}$ is the percent of the region that has been paved. For a particular rural region, $\mathrm{P}=1400$ when $x=0$ . Predict what the quail population will be in this region when it becomes $9 \%$ paved. Round your answer to the nearest whole number.

An accountant tabulated a firm's profits for four recent years in the following table: The accountant then fit both a linear graph and an exponential curve (seen below) to the data, in order to estimate future profits. Use the linear graph to estimate the profits in the year 2002.

The screen shows a table of values for the function defined by $Y_{1}=\left(0.95+\frac{1}{X}\right)^{X}$ . What number does the function value seem to approach as $x$ takes on larger and larger values?

Evaluate the given logarithm by applying the appropriate rule(s). DO NOT USE A CALCULATOR. -Given $\log _{10} 2=0.3010$ and $\log _{10} 3=0.4771$ , evaluate $\log _{10} 24$ .

Express the given logarithm as a sum and/or difference of logarithms. Simplify, if possible. Assume that all variables represent positive real numbers. - $\log _{n} \sqrt[3]{\frac{2 x^{6}}{z^{9}}}$

Find the logarithm. Give an approximation to four decimal places - $\ln \left(5.59 \times 10^{22}\right)$

Determine whether or not the function is one-to-one. - $f(x)=2 x-4$

Evaluate the logarithm. - $\log _{3}\left(\frac{1}{3}\right)$

Graph the given function as a solid line (or curve) and its inverse as a dashed line (or curve) on the same set of axes. - $f(x)=3 x$

Solve the problem. -The growth in the population of a certain rodent at a dump site can be modeled by the exponential function $A(t)=141 e^{0.027} t$ , where $t$ is the number of years since 1972. Estimate the population in the year 2000 .

Solve the equation. - $4^{X}=16$

Graph the given function as a solid line (or curve) and its inverse as a dashed line (or curve) on the same set of axes. - $f(x)=x^{3}+5$