Exam 4: Exponential and Logarithmic Functions

Solve the problem. -Daryl borrows $3,750 at a rate of 10.5% compounded semiannually. Find how much Daryl owes at the end of 5 years. Use: $A = P \left( 1 + \frac { r } { n } \right) ^ { n t }$ where: $\mathrm { A } =$ final amount $\mathrm { P } = \$ 3,750$ (the amount borrowed) $r = 10.5 \% = 0.105$ (the annual rate of interest) $\mathrm { n } = 2$ (the number of times interest is compounded each year) $t = 5$ (the duration of the loan in years)

Write as the sum and/or difference of logarithms. Express powers as factors. - $\log _ { 7 } \frac { 5 } { 9 }$

Choose the one alternative that best completes the statement or answers the question. -In a town whose population is 3000 , a disease creates an epidemic. The number of people, $N$ , infected $t$ days afte: disease has begun is given by the function $\mathrm { N } ( \mathrm { t } ) = \frac { 3000 } { 1 + 21.2 \mathrm { e } ^ { - 0.54 t } }$ . Find the number of infected people after 10 days.

Choose the one alternative that best completes the statement or answers the question. Graph the function using a graphing utility and the Change-of-Base Formula. - y=12x

Choose the one alternative that best completes the statement or answers the question. -To remodel a bathroom, a contractor charges $\$ 25$ per hour plus material costs, which amount to $\$ 3,775$ . Therefore, the total cost to remodel the bathroom is given by $f ( x ) = 25 x + 3,775$ where $x$ is the number of hours the contractor works. Find a formula for $\mathrm { f } ^ { - 1 } ( \mathrm { x } )$ . What does $\mathrm { f } ^ { - 1 } ( \mathrm { x } )$ compute?

Solve the equation. - $2 ^ { ( 7 + 3 x ) } = \frac { 1 } { 4 }$

Write the word or phrase that best completes each statement or answers the question. -A particular Boeing 747 jetliner produces noise at a loudness level of 113 decibels. Find the intensity level (round to the nearest hundredth) in watt per square meter for this noise.

Choose the one alternative that best completes the statement or answers the question. Find the domain of the composite function f ° g. - $f ( x ) = \frac { - 7 } { x } ; \quad g ( x ) = \frac { - 2 } { x + 1 }$

Choose the one alternative that best completes the statement or answers the question. Indicate whether the function is one-to-one. -{(5, 4), (6, 4), (7, 3), (8, -9)}

Write the word or phrase that best completes each statement or answers the question. -Instruments on a satellite measure the amount of power generated by the satellite's power supply. The time $t$ and the power $\mathrm { P }$ can be modeled by the function $\mathrm { P } = 50 \mathrm { e } ^ { - \mathrm { t } / 300 }$ , where $\mathrm { t }$ is in days and $\mathrm { P }$ is in watts. How much power will be available after 378 days? Round to the nearest hundredth.

Express as a single logarithm. - $28 \log _ { 7 } \sqrt [ 7 ] { x } + \log _ { 7 } \left( 28 x ^ { 6 } \right) - \log 728$

Find the present value. Round to the nearest cent. -To get $10,500 after7 years at 4% compounded annually

Choose the one alternative that best completes the statement or answers the question. -A fossilized leaf contains 18% of its normal amount of carbon 14. How old is the fossil (to the nearest year)? Use 5600 years as the half-life of carbon 14.

Write the word or phrase that best completes each statement or answers the question. -Bob, the incredible shrinking man, loses half of his height each day after he was exposed to a mysterious form of cosmic radiation. How many days before he is literally "knee-high to a grasshopper"? Assume that a grasshopper's knee is 4 millimeters high and that Bob is 2 meters tall. Round your answer to the nearest whole day. (1000 millimeters = 1 meter)

Find functions f and g so that f ° g = H. - $H ( x ) = \sqrt { \frac { 1 } { x - 2 } }$

Graph the function. - $f ( x ) = e ^ { x + 5 }$

Write the word or phrase that best completes each statement or answers the question. -A thermometer is taken from a room at $71 ^ { \circ } \mathrm { F }$ to the outdoors where the temperature is $14 ^ { \circ } \mathrm { F }$ . Determine what the reading on the thermometer will be after 5 minutes, if the reading drops to $45 ^ { \circ } \mathrm { F }$ after 1 minute. Assume the cooling follows Newton's Law of Cooling: $\mathrm { U } = \mathrm { T } + \left( \mathrm { U } _ { \mathrm { O } } - \mathrm { T } \right) \mathrm { e } ^ { \mathrm { kt } } .$ (Round your answer to two decimal places.)

Solve the equation. - $\left( \frac { 1 } { 3 } \right) ^ { 4 x + 5 } = 9 ^ { x - 2 }$

Choose the one alternative that best completes the statement or answers the question. Solve the equation. - $\log _ { 8 } x ^ { 2 } = 4$

Choose the one alternative that best completes the statement or answers the question. Solve the exponential equation. Use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. - $4 ^ { 6 x } = 4.8$