Exam 4: Inverse, Exponential, and Logarithmic Functions

Provide an appropriate response. -The graph of an exponential function with base a is given. Sketch the graph of g(x) = ax + 2. Give the domain and range of g. $f ( x ) = a ^ { x }$

Solve the equation. - $9 = \mathrm { b } ^ { 2 / 5 }$

An everyday activity is described. Keeping in mind that an inverse operation ʺundoesʺ what an operation does, describe each inverse activity. -setting a clock forward by one hour

Write the word or phrase that best completes each statement or answers the question. -In the formula $\mathrm { N } = \mathrm { Ie } \mathrm { kt } , \mathrm { N }$ is the number of items in terms of an initial population $\mathrm { I }$ at a given time $\mathrm { t }$ and $\mathrm { k }$ is a growth constant equal to the percent of growth per unit time. How long will it take for the population of a certain country to double if its annual growth rate is $7.5 \%$ ? Round to the nearest year.

Decide whether the given functions are inverses. -

Solve the equation and express the solution in exact form. - $\log 5 x = \log 3 + \log ( x - 4 )$

Solve the equation and express the solution in exact form. - $\log _ { 3 } x = \sqrt { \log 3 x }$

Find the value. Give an approximation to four decimal places. - $\ln 112 - \ln 13$

Find the value. Give an approximation to four decimal places. - $\ln 572 + \ln 37$

Write the word or phrase that best completes each statement or answers the question. - $\text { Between what two integers does the solution of } \log _ { 7 } x = 4.45 \text { lie? }$

Solve the following graphically. If necessary, round answers to the nearest thousandth. - $e ^ { x } - 10 = - 24 e ^ { - x }$

Solve the equation. - $e ^ { 3 x - 1 } = \left( e ^ { 2 } \right) ^ { - x }$

Provide an appropriate response. - $\text { Explain how the graph of } y = \left( \frac { 1 } { 2 } \right) ^ { x } - 1 \text { can be obtained from the graph of } y = 2 ^ { x }$

Use the graph of f to sketch a graph of the inverse of f using a dashed curve. -

Solve the problem. -The sales of a mature product (one which has passed its peak) will decline according to the function S(t) = S0e-at, where t is time in years since the peak sales. Find the sales of a product 21 Years after its peak sales if a = 0.15 and S0 = 28,100.

Solve the problem. -If an earthquake measured $5.9$ on the Richter scale, what was the approximate intensity of the earthquake in terms of $\mathrm { I } _ { 0 }$ ? Intensity on the Richter scale is $\log _ { 10 } \left( \mathrm { I } / \mathrm { I } _ { 0 } \right)$ .

Graph the function. Give the domain and range. - $f(x)=\log _{2} x$

Write the word or phrase that best completes each statement or answers the question. -Suppose the government wants to impose a tax on fossil fuels to reduce carbon emissions. The cost benefit is modeled by $\ln ( 1 - P ) = - 0.0034 - 0.0047 x$ , where $x$ represents the dollars of tax per ton of carbon emitted and $\mathrm { P }$ represents the percent reduction in emissions of carbon. ( $\mathrm { P }$ is in decimal form.) Determine $\mathrm { P }$ when $\mathrm { x } = 54$ . Round to three decimal places.

Solve the equation. - $\left( \frac { 1 } { 9 } \right) ^ { x } = 729$

Write the word or phrase that best completes each statement or answers the question. -The hydrogen potential, $\mathrm { pH }$ , of a substance is defined by $\mathrm { pH } = - \log \left[ \mathrm { H } ^ { + } \right]$ , where $\left[ \mathrm { H } ^ { + } \right]$ is measured in moles per liter. Find the pH of a sample of lake water whose $\left[ \mathrm { H } ^ { + } \right]$ is $3.05 \times 10 ^ { - 9 }$ moles per liter. (Round to the nearest tenth.)