Exam 4: Exponential and Logarithmic Functions

Consider the function $f ( x ) = e ^ { - ( x - 8 ) ^ { 2 } / 7 }$ . For what value of x does this function attain its maximum value, and what is the maximum function value? Round maximum function value to two decimal places, if necessary.

Solve for x. Give an exact answer and one rounded to two decimal places. 9^x = 94

It is projected that t years from now, the population of a certain country will be $P ( t ) = 60 e ^ { 0.06 t }$ million. What will be the population in 10 years? Round your answer to two decimal places, if necessary.

How quickly will $2,000 grow to $10,000 if interest is 7% compounded quarterly? Round your answer to the nearest whole year, if necessary.

Solve for x. Round to three decimal places, if necessary. $\log _ { 6 } x = 4$

Solve for x: $\log _ { 64 } x = \frac { 1 } { 3 }$

How much money should be invested today at 5 percent compounded continuously so that 10 years from now it will be worth $10,000?

How much money should be invested today at an annual interest rate of 7% compounded continuously so that 40 years from now it will be worth $5,000?

The gross national product (GNP) of a certain country was 150 billion dollars in 1988 and 160 billion in 1992. Assuming that the GNP grows exponentially, how much does it grow (in billions of dollars) between 1992 and the year 2000? Round your answer to two decimal places.

The population density x miles from the center of a certain city is $D ( x ) = 9 e ^ { - 0.06 x }$ thousand people per square mile. What is the population density 7 miles from the center of this city? Round your answer to two decimal places.

Differentiate the given function. $f ( x ) = \ln x ^ { 6 }$

Evaluate the expression $e ^ { 2 \ln 3 - 3 \ln 4 }$ without using tables or a calculator.

Solve for x: $\log _ { 5 } ( x - 1 ) = 3$

Solve for x: $2 ^ { x } = e ^ { 3 }$

Solve for x: $6 = 8 - 3 e ^ { - 8 x }$

If ln x = 3(ln 2 - ln 7), then $x = \frac { 1 } { 6 }$ .

The fraction of television sets manufactured by a certain company that are still in working condition after t years of use is approximately $f ( t ) = e ^ { - 0.1 t }$ . What fraction can be expected to fail before 5 years of use? Round your answer to two decimal places, if necessary.

The equation of the tangent line to f (x) = ln x at x = 1 is

The equation of the tangent line to $f ( x ) = 4 e ^ { - x }$ at x = 0 is y = 4x + 4.

If $7,000 is invested at 6% interest compounded continuously, the balance after 10 years will be $12,725.31.