Exam 4: Exponential and Logarithmic Functions

If log₇ (2x) = 2 , then $x = \frac { 49 } { 2 }$ .

Sally needs $2,000 for a trip she plans to take 3 years from now. How much, in dollars, should she invest in an account that pays 7% interest compounded continuously in order to meet her goal?

Solve the given equation for x. $7 = 6 + 4 e ^ { - 5 x }$

Let $f ( t ) = \frac { 13 } { 3 + 6 e ^ { - t / 140 } }$ . Which value of t corresponds to a possible inflection point forf (t)?

A sum of money, $A _ { 0 }$ , is invested at a certain fixed interest rate, and this interest is compounded continuously. After 6 years, the money has doubled. The balance at the end of 12 years is $2 A _ { 0 }$ .

A radioactive substance decays exponentially. If 800 grams were present initially and 600 grams are present 100 years later, how many grams will be present after 400 years? Round your answer to two decimal places.

The function f (x) = ln x is concave downward everywhere.

Solve for x. Give an exact answer and then round to two decimal places, if necessary. $7 ^ { - 8 x + 2 } = 85$

Simplify: $\left( 81 x ^ { 7 } y ^ { 10 } \right) ^ { 1 / 2 }$

It is estimated that t years from now, the population of a certain country will be $P ( t ) = \frac { 200 } { 1 + 9 e ^ { - 0.02 t } }$ million. When will the population be growing most rapidly? Round your answer to two decimal places.

Simplify: $\frac { 4 ^ { 5 } } { 4 ^ { 4 } }$

Suppose the number of bacteria grows from 7,000 to 10,000 in the first 15 minutes of an experiment. Assuming that the number of bacteria grows exponentially, how many bacteria will be present after 1 hour? Round to the nearest whole number.

Evaluate the given expression. $\left( \frac { 59,049 } { 100,000 } \right) ^ { 6 / 5 }$

How much money, in dollars, should be invested today at 7 percent compounded continuously so that 5 years from now it will be worth $5,000?

Solve the given equation for x. $- 2 = - 3 + 4 e ^ { - 5 x }$

Differentiate the given function. $f ( x ) = \frac { x ^ { 4 } } { e ^ { 2 x } }$

Simplify $\frac { 9 ^ { 3 } } { 3 ^ { 2 } }$ .

Simplify. $\left( 4 x ^ { 2 } y ^ { - 6 } z ^ { 3 } \right) ^ { - 2 }$ Your answer should include only positive exponents.

The equation of the tangent line to $f ( x ) = 7 e ^ { x }$ at x = 0 is y = 7x + 7.

Determine the monthly car payment for a new car costing $19,723, if there is a down payment of $3,000 and the car is financed over a 10-year period at an annual rate of 8% compounded monthly.