Exam 5: Exponential and Logarithmic Functions

The sales of a mature product (one which has passed its peak) will decline by the function $S _ { ( t ) } = S _ { 0 } e ^ { - a t }$ , where $t$ $s$ time in years. Find the sales after 16 years if a $= 0.19$ and $S _ { 0 } = 81,600$ .

Graph the function. - $f ( x ) = 2 \cdot 0.7 ^ { 4 x }$

The number of students infected with the flu on a college campus after t days is modeled by the function $\mathrm { P } ( \mathrm { t } ) = \frac { 600 } { 1 + 39 \mathrm { e } ^ { - 0.3 \mathrm { t } } }$ What was the initial number of infected students?

Wind speed varies in the first twenty meters above the ground. For a particular day, let $f ( x ) = 9.8 \ln x + 3.3$ model the wind speed $x$ meters above the ground. At what height is the wind speed 9 meters per second? Round results to the nearest hundredth.

Find the exponential function f that models this data. Round the coefficients to the nearest hundredth. x 3 6 9 12 y 12.2 36.2 73.5 151

Find the amount of money in an account after 5 years if $2800 is deposited at 7% annual interest compounded monthly.

Find the exponential function that satisfies the given conditions. -Initial mass $= 7 \mathrm {~g}$ , decreasing at a rate of $3.7 \%$ per day

A couple who wants to purchase a home with a price of $325,000 has $100,000 for a down payment. If they can get a 25-year mortgage at 8% per year, paid on the unpaid balance, what is the total amount they will pay Before they own the house outright? How much interest will they pay over the life of the loan?

Match the equation with its graph. - $y = 4 ( x + 4 ) + 1$

Provide an appropriate response. -Consider the logistic function $f ( x ) = \frac { c } { 1 + a e ^ { - b x } } , \text { where } b > 0$ . What is the limiting value of this function?

The natural resources of an island limit the growth of the population. The population of the island is given by the logistic equation $\mathrm { P } ( \mathrm { t } ) = \frac { 4943 } { 1 + 3.96 \mathrm { e } ^ { - 0.31 t } }$ , where t is the number of years after 2010. What is the limiting value Of the population?

Graph the function. - $f(x)=\left(\frac{1}{4}\right)^{x}$

Rewrite as a single logarithm. - $2 \log _ { 2 } ( 2 x + 5 ) + 6 \log _ { 2 } ( 6 x + 1 )$

An economist predicts that the buying power B(x) of a dollar x years from now will decrease according to the form $B ( x ) = 0.61 ^ { x }$ . How much will today's dollar be worth in 7 years? Round the answer to the nearest cent.

Find the inverse of the function. - $f ( x ) = 9 ^ { x } + 3$

Explain how the graph of $y = - 5 \left( \frac { 1 } { 2 } \right) ^ { x } \text { can be obtained from the graph of } y = 2 ^ { x }$

Assume the cost of a car is $20,000. With continuous compounding in effect, the cost of the car will increase according to the equation C $C = 20,000 \mathrm { e } ^ { \mathrm { rt } }$ , where r is the annual inflation rate and t is the number of years. Find The number of years it would take to double the cost of the car at an annual inflation rate of 5.9%. Round the Answer to the nearest hundredth.

The sales of a mature product (one which has passed its peak) will decline by the function $\left. \mathrm { S } ^ { \mathrm { t } } \right) = \mathrm { S } _ { 0 } \mathrm { e } ^ { - a t }$ , where $\mathrm { t }$ is time in years. Find the sales after 8 years if $\mathrm { a } = 0.18$ and $\mathrm { S } _ { 0 } = 48,500$ .

Provide an appropriate response. -What are the domain and range for the equation $y = 2 ^ { x }$ ?

Determine whether or not the given function is an exponential function. - $y = 4 ^ { x }$