Exam 4: Exponential and Logarithmic Functions

Differentiate the given function. $f ( x ) = \ln x ^ { 5 }$

$f ( x ) = \frac { \ln x } { e ^ { x } }$ , then $f ^ { \prime } ( x ) = \frac { 1 / x - \ln x } { e ^ { x } }$ .

Find the effective interest rate if the nominal rate is 7 percent compounded continuously. Round to two decimal places, if necessary.

Use logarithm rules to rewrite the expression in terms of log x and log y. $\log \left( x ^ { 1 / 3 } y ^ { 4 } \right)$

The total number of hamburgers sold by a fast food chain is growing exponentially. If 3 billion have been sold by 1998 and 9 billion by 2000, how many will be sold in the year 2010? Round your answer to one decimal place.

The population density x miles from the center of a certain city is $D ( x ) = 10 e ^ { - 0.04 x }$ thousand people per square mile. What is the population density 9 miles from the center of this city? Round your answer to two decimal places.

Solve for x: $a ^ { 7 x - 1 } = b$

Solve the given equation for x. $10 = 9 + 8 e ^ { 9 x }$

It takes 8 years for money to double with quarterly compounding if the annual interest rate is 8%.

How quickly will $700 grow to $2,000 if interest is 8% compounded quarterly? Round your answer to two decimal places, if necessary.

Suppose your family owns a rare book whose value t years from now will be $V ( t ) = 8 e ^ { \sqrt { 0.6 t } }$ dollars. If the prevailing interest rate remains constant at 6% per year compounded continuously, when will it be most advantageous for your family to sell the book and invest the proceeds? Round your answer to two decimal places.

Solve for x. Round to three decimal places, if necessary. $\log _ { 2 } x = 3$

Find the derivative of $\ln \left[ \left( \ln x ^ { 4 } \right) ^ { 3 } \right]$ .

If $3,000 is invested at 10% compounded continuously, what is the balance after 9 years?

If $y = e ^ { - 3 x } \ln \sqrt { x }$ , then $\frac { d y } { d x } = - 3 e ^ { - 3 x } \ln \sqrt { x }$ .

Simplify. $\left( 2 x ^ { 9 } y ^ { - 8 } \right) \left( 4 x ^ { - 12 } y ^ { 9 } \right)$ Your answer should include only positive exponents.

Records indicate that t weeks after the outbreak of a disease, approximately $Q ( t ) = \frac { 70 } { 3 + 52 e ^ { - 13 t } }$ thousand people have been infected. At what rate was the disease spreading at the end of the third week? Round your answer to two decimal places.

If $f ( x ) = \left( x + 7 e ^ { - x } \right) ^ { 8 }$ , then $f ^ { \prime } ( x ) = x + 7 e ^ { - x }$ .

Solve for x to two decimal places: $\ln \sqrt { x } = \frac { 1 } { 5 } ( \ln 8 + 2 \ln 3 )$

A traffic accident was witnessed by $\frac { 1 } { 12 }$ of the residents of a small town. The number of residents who had heard about the accident t hours later is given by a function of the form $\frac { B } { 1 + C e ^ { - k t } }$ , where B is the population of the town. If $\frac { 1 } { 6 }$ of the residents had heard about the accident after 1 hours, how long did it take for $\frac { 1 } { 3 }$ of the residents to hear the news? Round your answer to two decimal places.