Exam 4: Exponential and Logarithmic Functions

Choose the one alternative that best completes the statement or answers the question. -A nuclear scientist has a sample of $100 \mathrm { mg }$ of a radioactive material which has a half-life in hours. She monitors the amount of radioactive material over a period of a day and obtains the following data. Use a graphing utility to fit an exponential function to the data. Predict the amount of material remaining at 40 hours. Hours 0 5 10 15 20 25 30 100 68.3 45.2 31.3 21.5 14.6 9.8

Write as the sum and/or difference of logarithms. Express powers as factors. - $\log _ { 8 } \frac { 11 \sqrt { \mathrm { r } } } { \mathrm { s } }$

Solve the problem. -An oil well off the Gulf Coast is leaking, with the leak spreading oil over the surface of the gulf as a circle. At any time t, in minutes, after the beginning of the leak, the radius of the oil slick on the surface is r(t) = 3t ft. Find The area A of the oil slick as a function of time.

Find functions f and g so that f ° g = H. - $H ( x ) = \frac { 1 } { x ^ { 2 } - 6 }$

The graph of an exponential function is given. Match the graph to one of the following functions. -

Solve the exponential equation. Express the solution set in terms of natural logarithms. - $4 ^ { x + 6 } = 7$

Choose the one alternative that best completes the statement or answers the question. Solve the equation. Express irrational answers in exact form and as a decimal rounded to 3 decimal places. - $\left( \frac { 1 } { 2 } \right) ^ { x } = 7 ^ { 1 - x }$

Write the word or phrase that best completes each statement or answers the question. -The size $\mathrm { P }$ of a certain insect population at time $\mathrm { t }$ (in days) obeys the function $\mathrm { P } = 700 \mathrm { e } ^ { 0.03 \mathrm { t } }$ . After how many days will the population reach 1500 ?

Solve the equation. - $\log _ { 21 } ( x + 84 ) = 3 - \log _ { 21 } x$

Write the word or phrase that best completes each statement or answers the question. Use transformations to graph the function. Determine the domain, range, and vertical asymptote of the function. - $h(x)=3+\log (x+4)$

Solve the problem. -A city is growing at the rate of 0.5% annually. If there were 3,110,000 residents in the city in 1,995, find how many (to the nearest ten-thousand) were living in that city in 2000. Use y = 3,110,000(2.7)0.005t

Choose the one alternative that best completes the statement or answers the question. Find the domain of the composite function f ° g. - $f ( x ) = x + 8 ; \quad g ( x ) = \frac { 4 } { x + 6 }$

Decide whether the composite functions, f ° g and g ° f, are equal to x. - $f ( x ) = \sqrt { x } , g ( x ) = x ^ { 2 }$

Write the word or phrase that best completes each statement or answers the question. -Gillian has $10,000 to invest in a mutual fund. The average annual rate of return for the past five years was 12.25%. Assuming this rate, determine how long it will take for her investment to double.

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

Write the word or phrase that best completes each statement or answers the question. -A music store manager collected data regarding price and quantity demanded of cassette tapes every week for 10 weeks, and found that the exponential function of best fit to the data was $p = 25 \cdot 0.89 q$ . Express the function of best fit in the form $\mathrm { p } = \mathrm { p } _ { 0 } \mathrm { e } ^ { \mathrm { kq } }$ , and use this expression to predict the quantity demanded if the price is $\$ 8.50$ .

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. - $\mathrm { e } ^ { 2 \mathrm { x } } = 4$

Find functions f and g so that f ° g = H. - $H ( x ) = \left| 4 - 3 x ^ { 2 } \right|$

Find the value of the expression. -Let $\log _ { b } A = 4$ and $\log _ { b } B = - 5$ . Find $\log _ { b } 2 \sqrt { A B }$ .

Express as a single logarithm. - $\log _ { c } m + \log _ { c } n$