Q: A population of bacteria triples every $$5$$ hours. If the initial population is $$200$$, what is the population after $$15$$ hours? (Assume exponential growth: $$P(t) = P_0$ e$$^{kt}$$.)

$$P(15) = 200 \times 3$$^{3}$$ = 200 \times 27 = 5400$$

Q: John invests $$5000 in an $$account that earns $$3.25\%$$ annual interest, compounded continuously. What is the value of the account after $$5$$ years? (Use $$A = Pe$$^{rt}$$.)

$$A = 5000 \times e$$^{0.0325 \times 5}$$ \approx 5000 \times e$$^{0.1625}$$ \approx 5000 \times 1.1764 \approx 5882$$

Q: A medication decays in the bloodstream according to $$C(t) = 50e$$^{-0.2t}$$, where C$ is the concentration in mg/L and t$ is in hours. What is the concentration after 8$ hours?

$$C(8) = 50e$$^{-0.2 \times 8}$$ = 50e$$^{-1.6}$$ \approx 50 \times 0.2019 \approx 10.10 mg/L$$

Q: A radioactive substance decays according to $$Q(t) = Q_0$ e$$^{-kt}$$. If Q_0$ = 100$$ grams and $$k = 0.03$$, how much remains after $$10$$ years?

$$Q(10) = 100e$$^{-0.3}$$ \approx 100 \times 0.7408 \approx 74.08$$ grams

Question 1

Suppose $$P = 8000 is $$invested at an annual interest rate of $$4.5\%$$ compounded quarterly. What is the value of the investment after $$6$$ years? Use the formula $$A = P \left(1 + \frac{r}{n}\$$right)^{nt}$$.

Accepted Answer

$$A = 8000 \left(1 + \frac{0.045}{4}\$$right)^{24}$$ \approx 10413.60$$

Question 2

A population of bacteria triples every $$5$$ hours. If the initial population is $$200$$, what is the population after $$15$$ hours? (Assume exponential growth: $$P(t) = P_0$ e$$^{kt}$$.)

Accepted Answer

$$P(15) = 200 	imes 3$$^{3}$$ = 200 	imes 27 = 5400$$

Question 3

Which of the following is the correct formula for the amount $$A$$ after $$t$$ years if $$P is $$invested at an annual rate $$r$$ compounded continuously?

Accepted Answer

$$A = Pe$$^{rt}$$

Question 4

John invests $$5000 in an $$account that earns $$3.25\%$$ annual interest, compounded continuously. What is the value of the account after $$5$$ years? (Use $$A = Pe$$^{rt}$$.)

Accepted Answer

$$A = 5000 	imes e$$^{0.0325 	imes 5}$$ \approx 5000 	imes e$$^{0.1625}$$ \approx 5000 	imes 1.1764 \approx 5882$$

Question 5

Marisa invests $$12,000 in an $$account that earns $$2.8\%$$ annual interest, compounded continuously. When will Marisa’s account double in value? (Solve for $$t in$$ $$A = Pe$$^{rt}$$.)

Accepted Answer

$$t = \frac{\ln 2}{0.028} \approx 24.75$$ years

Question 6

Which of the following properties of logarithms is correct?

Accepted Answer

$$\ln(MN) = \ln M + \ln N$$

Question 7

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

Accepted Answer

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

Question 8

Which of the following statements about the natural logarithm function $$\ln x is $$true?

Accepted Answer

$$\ln x is $$only defined for $$x \u003e 0$$

Question 9

If $$A = 10,000$$ and $$A = Pe$$^{0.05t}$$, how long will it take for the investment to triple?

Accepted Answer

$$t = \frac{\ln 3}{0.05} \approx 21.97$$ years

Question 10

Which of the following is the correct inverse of the function $$f(x) = e$$^{3x}$$?

Accepted Answer

$$f^{-1}(x$$) = \frac{1}{3} \ln x$$

Question 11

A medication decays in the bloodstream according to $$C(t) = 50e$$^{-0.2t}$$, where C$ is the concentration in mg/L and t$ is in hours. What is the concentration after 8$ hours?

Accepted Answer

$$C(8) = 50e$$^{-0.2 	imes 8}$$ = 50e$$^{-1.6}$$ \approx 50 	imes 0.2019 \approx 10.10 mg/L$$

Question 12

Which of the following is the correct domain of the function $$f(x) = \ln(x-3)$$?

Accepted Answer

$$x \u003e 3$$

Question 13

If $$f(x) = e$$^{-x}$$, what is the horizontal asymptote of f(x$$)$$?

Accepted Answer

$$y = 0$$

Question 14

Solve for $$x$$: $$\ln(x) = 4.$$

Accepted Answer

$$x = e$$^{4}$$

Question 15

A radioactive substance decays according to $$Q(t) = Q_0$ e$$^{-kt}$$. If Q_0$ = 100$$ grams and $$k = 0.03$$, how much remains after $$10$$ years?

Accepted Answer

$$Q(10) = 100e$$^{-0.3}$$ \approx 100 	imes 0.7408 \approx 74.08$$ grams

Question 16

Which of the following is equivalent to $$\ln\left(\frac{a}{b}\right)$$?

Accepted Answer

$$\ln a - \ln b$$

Question 17

If $$f(x) = \ln(x^2$ + 1)$$, what is the domain of $$f(x)$$?

Accepted Answer

All real numbers

Question 18

A savings account offers $$2.5\%$$ annual interest, compounded monthly. If you deposit $$P = 3000$$, what is the value after $$4$$ years? (Use $$A = P(1 + \frac{r}{n}$$)^{nt}$$.)

Accepted Answer

$$A = 3000 \left(1 + \frac{0.025}{12}\$$right)^{48}$$ \approx 3000 	imes 1.104 \approx 3312$$

Question 19

Which of the following is the correct property of exponents for $$e^{a}$$ \cdot $$e^{b}$$?

Accepted Answer

$$e^{a}$$ \cdot $$e^{b}$$ = $$e^{a+b}$$

Exponential and Logarithmic Functions in Business Calculus

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