Q: Suppose you invest \$$10,000 in an account that earns 5\% annual interest compounded monthly. What will the balance be after 8 years? Use the formula A = P$$ \left(1 + \frac{r}{n}\$$right)^{nt}$.

$$A = 10,000 \left(1 + \frac{0.05}{12}\$$right)^{12 \times 8}$$ \approx 14,859$$

Question 1

Suppose you invest \$$10,000 in an account that earns 5\% annual interest compounded monthly. What will the balance be after 8 years? Use the formula A = P$$ \left(1 + \frac{r}{n}\$$right)^{nt}$.

Accepted Answer

$$A = 10,000 \left(1 + \frac{0.05}{12}\$$right)^{12 	imes 8}$$ \approx 14,859$$

Question 2

Solve for $$x$$: $$e^{2x} = 7$.

Accepted Answer

$$x = \frac{1}{2} \ln 7$$

Question 3

A radioactive substance decays according to $$A = A_0$ e$$^{-kt}$$. If a sample has a half-life of 20 years, what is the value of k$?

Accepted Answer

$$k = \frac{\ln 2}{20}$$

Question 4

Which of the following is equivalent to $$\ln(e$$^{3x}$$)$$?

Accepted Answer

$$3x$$

Question 5

If $$\log_{b} 8 = 3$$, what is the value of $$b$$?

Accepted Answer

$$b = 2$$

Question 6

A population grows exponentially according to $$P(t) = 5000 e$$^{0.03t}$$. What is the population after 10 years?

Accepted Answer

$$P(10) = 5000 e$$^{0.3}$$ \approx 6749$$

Question 7

Solve for $$x$$: $$\ln(x - 2) = 3.$$

Accepted Answer

$$x = e$$^{3}$$ + 2$$

Question 8

Which of the following is the solution to $$2^{x} = 10$?

Accepted Answer

$$x = \log_{2} 10$$

Question 9

If $$A = P e$$^{rt}$$, how long will it take for an investment to double if the interest rate is 6\% compounded continuously?

Accepted Answer

$$t = \frac{\ln 2}{0.06}$$

Question 10

Which of the following is the inverse function of $$f(x) = e$$^{x}$$?

Accepted Answer

$$f^{-1}(x$$) = \ln x$$

Question 11

A medication is eliminated from the bloodstream at a rate proportional to the amount present. If the half-life is 4 hours, what fraction remains after 12 hours?

Accepted Answer

$$\left(\frac{1}{2}\$$right)^{3}$$ = \frac{1}{8}$$

Question 12

If $$\log_{10} x = 2$$, what is $$x$$?

Accepted Answer

$$x = 100$$

Question 13

Which of the following expressions is equivalent to $$\log_{a}(xy)$$?

Accepted Answer

$$\log_{a} x + \log_{a} y$$

Question 14

A car depreciates at a rate of 12\% per year. If the car is currently worth \$$20,000, what will it be worth in 5 years? Use A = P(1$$ - $$r)^{t}$.

Accepted Answer

$$A = 20,000 (0.88$$)^{5}$$ \approx 10,747$$

Question 15

Solve for $$x$$: $$3^{2x-1} = 27$.

Accepted Answer

$$x = 2$$

Question 16

Which of the following is the correct formula for the amount $$A$$ after $$t$$ years for an initial principal $$P at an $$annual interest rate $$r$$ compounded $$n$$ times per year?

Accepted Answer

$$A = P \left(1 + \frac{r}{n}\$$right)^{nt}$$

Question 17

If $$\log_{b} 81 = 4$$, what is $$b$$?

Accepted Answer

$$b = 3$$

Question 18

A scientist starts with 100 grams of a substance that decays at a rate of 7\% per year. How much remains after 10 years? Use $$A = P(1 - r$$)^{t}$$.

Accepted Answer

$$A = 100 (0.93$$)^{10}$$ \approx 48.1$$

Question 19

Which of the following is equivalent to $$\log_{a} \left(\frac{x}{y}\right)$$?

Accepted Answer

$$\log_{a} x - \log_{a} y$$

Question 20

If $$f(x) = \ln(x)$$, what is the domain of $$f$$?

Accepted Answer

$$(0, \infty)$$

Exponential and Logarithmic Functions: Applications in Finance and Growth

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