Exam 5: Inverse, Exponential, and Logarithmic Functions

Graph the exponential function using transformations where appropriate. - $f ( x ) = 2 ^ { x } + 2$

The decay of 921 mg of an isotope is given by A(t) = 921e-0.031t, where t is time in years. Find the amount left after 41 years.

Find the value. Give an approximation to four decimal places. - $\ln 838 + \ln 30$

Solve the equation. - $a ^ { 3 / 5 } = 8$

Suppose the government wants to impose a tax on fossil fuels to reduce carbon emissions. The cost benefit is modeled by $\ln ( 1 - \mathrm { P } ) = - 0.0031 - 0.0045 \mathrm { x }$ , where $x$ represents the dollars of tax per ton of carbon emitted and $\mathrm { P }$ represents the percent reduction in emissions of carbon. ( $P$ is in decimal form.) Determine $P$ when $x = 44$ . Round to three decimal places.

Decide whether the pair of functions graphed are inverses. -

Explain why $\log _ { a } 5 = 5$

Solve the equation. - $e ^ { x - 4 } = \left( \frac { 1 } { e ^ { 3 } } \right) ^ { x + 4 }$

Use the formu $\mathrm { P } = \mathrm { Ie } ^{ kt }$ . A bacterial culture has an initial population of 10,000. If its population declines to 6000 in 6 hours, what will it be at the end of 8 hours?

Use the change of base rule to find the logarithm to four decimal places. - $\log \sqrt { 6 } 118.7$

Use the definition of inverses to determine whether f and g are inverses. - $f ( x ) = \sqrt { x + 5 } , x \geq - 5 ; \quad g ( x ) = x ^ { 2 } + 5$

The purchasing power of a dollar is decreasing at the rate of 4.9% annually, compounded continuously. How long will it take for the purchasing power of $1.00 to be worth $0.74? Round answers to the nearest hundredth.

The graph of $y = e ^ { x }$ is shown with the coordinates of a point displayed at the bottom of the screen. Write the logarithmic equation associated with the display.

The height in meters of women of a certain tribe is approximated by $h = 0.52 + 2 \log ( t / 3 )$ where $t$ is the woman's age in years and $1 \leq t \leq 20$ . Estimate the height (to the nearest hundredth) of a woman of the tribe 4 years of age.

Solve the equation. - $e ^ { 2 x - 1 } = \left( e ^ { 6 } \right) ^ { - x }$

A population is increasing according to the exponential function defined by $\mathrm { y } = 3 \mathrm { e } .04 \mathrm { x }$ , where $\mathrm { y }$ is in millions and $x$ is the number of years. Which of the following should be done in order to answer the question "How large will the population be in 4 years?"

Suppose th $y = \frac { 2 - \log ( 100 - x ) } { 0.30 }$ can be used to calculate the number of years y for x percent of a population of 759 web-footed sparrows to die. Approximate the percentage (to the nearest whole per cent) of web-footed Sparrows that died after 3 years.

What is the rate on an investment that doubles $2520 in 17 years? Assume interest is compounded quarterly.

Explain the error in the following: $\log _ { 4 } 3 y = \log _ { 4 } 3 \cdot \log _ { 4 } y$

Decide whether the given functions are inverses. -