Q: Which of the following best defines the natural base $$e$$?

$$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$$

Q: A company invests $$P = \10$$,000$$ at a continuous annual rate of r = 5$$\%$$ for t = 4$ years. What is the future value A$ of the investment? (Round to two decimal places)

$$A = 10,000e$$^{0.05 \times 4}$$ = \12$$,214.03$

Q: A medication degrades continuously at a rate of $$r = 0.08$$ per hour. If the initial amount is $$P = 100 mg$$, how much remains after $$t = 5$$ hours?

$$A = 100e$$^{-0.08 \times 5}$$ \approx 67.03 mg$$

Q: A scientist finds that a bacteria culture grows continuously at a rate of $$r = 0.09$$ per hour. If the initial population is $$P = 500$$, how many bacteria will there be after $$t = 10$$ hours?

$$A = 500e$$^{0.09 \times 10}$$ \approx 1,227$$

Question 1

Which of the following best defines the natural base $$e$$?

Accepted Answer

$$e = \lim_{n 	o \infty} \left(1 + \frac{1}{n}ight)^n$$

Question 2

A company invests $$P = \10$$,000$$ at a continuous annual rate of r = 5$$\%$$ for t = 4$ years. What is the future value A$ of the investment? (Round to two decimal places)

Accepted Answer

$$A = 10,000e$$^{0.05 	imes 4}$$ = \12$$,214.03$

Question 3

Which property of natural logarithms allows you to write $$\ln(MN) as$$ $$\ln M + \ln N$$?

Accepted Answer

Product Property

Question 4

If $$f(x) = \ln(7 - 3x)$$, what is the domain of $$f$$?

Accepted Answer

$$(-\infty, \frac{7}{3})$$

Question 5

Solve for $$x$$: $$5 + 7\ln(x + 3) = 19$$

Accepted Answer

$$x \approx 4.389$$

Question 6

Which of the following is equivalent to $$y = \ln x$$?

Accepted Answer

$$e^y = x$$

Question 7

Given $$\ln 2 = 0.6931$$ and $$\ln 3 = 1.0986$$, calculate $$\ln(2 \cdot 3).$$

Accepted Answer

$$1.7917$$

Question 8

If a population doubles every 6 years, what is the continuous exponential growth rate $$r$$? (Round to three decimal places)

Accepted Answer

$$r = \frac{\ln 2}{6} \approx 0.116$$

Question 9

Which of the following is NOT a property of the natural logarithm?

Accepted Answer

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

Question 10

A medication degrades continuously at a rate of $$r = 0.08$$ per hour. If the initial amount is $$P = 100 mg$$, how much remains after $$t = 5$$ hours?

Accepted Answer

$$A = 100e$$^{-0.08 	imes 5}$$ \approx 67.03 mg$$

Question 11

Which of the following statements about the function $$f(x) = \ln x is $$true?

Accepted Answer

Its domain is $$(0, \infty)$$ and its range is $$(-\infty, \infty).$$

Question 12

Solve for $$t$$: $$500e^{0.015t} = 750$

Accepted Answer

$$t \approx 40.55$$

Question 13

Which of the following is the correct expression for the future value $$A of an $$investment $$P$$ compounded continuously at rate $$r$$ for time $$t$$?

Accepted Answer

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

Question 14

If $$y = \ln e^4$, what is the value of y$?

Accepted Answer

$$4$$

Question 15

Which of the following is the correct way to rewrite $$\ln\left(\frac{2e^5$}{3}\right)$$ using properties of logarithms?

Accepted Answer

$$\ln 2 + 5 - \ln 3$$

Question 16

A startup’s user base grows continuously at a rate of $$r = 0.12$$ per month. How long will it take for the user base to triple?

Accepted Answer

$$t = \frac{\ln 3}{0.12} \approx 9.16$$ months

Question 17

Which of the following is true about the inverse relationship between $$e^x$$ and $$\ln x$$?

Accepted Answer

$$e^{\ln x} = x$ for all x$$ \u003e 0$$

Question 18

If $$\ln 2 = 0.6931$$, what is $$\ln 2^5$?

Accepted Answer

$$5 	imes 0.6931 = 3.4655$$

Question 19

A scientist finds that a bacteria culture grows continuously at a rate of $$r = 0.09$$ per hour. If the initial population is $$P = 500$$, how many bacteria will there be after $$t = 10$$ hours?

Accepted Answer

$$A = 500e$$^{0.09 	imes 10}$$ \approx 1,227$$

Exponential and Logarithmic Functions of the Natural Base $e$

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