Exam 4: Exponential and Logarithmic Functions

Write the word or phrase that best completes each statement or answers the question. -The half-life of carbon-14 is 5700 years. Find the age of a sample in which 8% of the radioactive nuclei originally present have decayed.

Find a formula for the inverse of the function described below. -An organization determines that the cost per person of chartering a bus is given by the formula $C ( x ) = \frac { 200 + 6 x } { x } \text {, }$ where $x$ is the number of people in the group and $C ( x )$ is in dollars.

For the given functions f and g, find the requested composite function. - $f ( x ) = - 2 x + 3 , \quad g ( x ) = 6 x + 4 ; \quad$ Find $( g \circ f ) ( x )$

Use the properties of logarithms to find the exact value of the expression. Do not use a calculator. - $\log _ { 30 } 10 + \log _ { 30 } 3$

Solve the problem. -Suppose that $f ( x ) = 2 ^ { x }$ . What is $f ( 3 )$ ? What point is on the graph of $f$ ?

Solve the equation. - $( 8 ) ) ^ { x + 1 } = \left( \frac { 2 } { 4 } \right) ^ { x - 1 }$

Graph the function. - $f ( x ) = 4 - e ^ { - x }$

Decide whether or not the functions are inverses of each other. - $f ( x ) = ( x - 2 ) ^ { 2 } , x \geq 2 ; g ( x ) = \sqrt { x } + 2$

For the given functions f and g, find the requested composite function value. - $f ( x ) = 2 x + 2 , \quad g ( x ) = 2 x ^ { 2 } + 3 ; \quad \text { Find } ( f \circ f ) ( 0 )$

Write the word or phrase that best completes each statement or answers the question. Solve the problem. -The profit P for selling x items is given by the equation P(x) . Express the sales amount x as a function $P ( x ) = 2 x - 500$ of the profit P.

Solve the problem. -The number of books in a small library increases according to the function B = 2,800e0.03t, where t is measured in years. How many books will the library have after 10 years?

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

Change the exponential expression to an equivalent expression involving a logarithm. - $32 ^ { 2 / 5 } = 4$

Write the word or phrase that best completes each statement or answers the question. -A rumor is spread at an elementary school with 1200 students according to the model N = 1200(1 - e-0.16d) where N is the number of students who have heard the rumor and d is the number of days that have elapsed since the rumor began. How many days must elapse for 500 to have heard the rumor?

Choose the one alternative that best completes the statement or answers the question. Find the effective rate of interest. -25.12% compounded daily

Solve the problem. Round your answer to three decimals. -How long will it take for an investment to triple in value if it earns 8.5% compounded continuously?

Use a calculator to find the natural logarithm correct to four decimal places. - $\ln 0.000356$

The function f is one-to-one. Find its inverse. - $f ( x ) = x ^ { 3 } + 4$

Solve the problem. -The $\mathrm { pH }$ of a chemical solution is given by the formula $\mathrm { pH } = - \log _ { 10 } \left[ \mathrm { H } ^ { + } \right]$ where $\left[ \mathrm { H } ^ { + } \right]$ is the concentration of hydrogen ions in moles per liter. Find the $\mathrm { pH }$ if the $\left[ \mathrm { H } ^ { + } \right] = 8.4 \times 10 ^ { - 1 }$ .

Choose the one alternative that best completes the statement or answers the question. -The function $\mathrm { A } = \mathrm { A } _ { \mathrm { O } } \mathrm { e } ^ { - 0.01386 \mathrm { x } }$ models the amount in pounds of a particular radioactive material stored in a concrete vault, where $x$ is the number of years since the material was put into the vault. If 600 pounds of the material are placed in the vault, how much time will need to pass for only 99 pounds to remain?