Q: A $$\10$$,000$$ investment is compounded continuously at an annual rate of 5$$\%$$. What will the balance be after 8 years? (Use A$$ = $$Pe^{rt}$)

$$A = 10,000 \cdot e$$^{0.05 \times 8}$$

Q: A radioactive substance decays according to $$A = A_0$ e$$^{-0.03t}$$. If the initial amount is 100$ grams, how much remains after 20$ years?

$$A = 100 \cdot e$$^{-0.03 \times 20}$$

Question 1

A biologist is studying the growth of a bacteria population. The population doubles every 3 hours. If the initial population is 500, which equation models the population $$P$$ after $$t$$ hours?

Accepted Answer

$$P = 500 \cdot 2$$^{t/3}$$

Question 2

Which of the following is the domain of the function $$f(x) = \log_{3}(x-2)$$?

Accepted Answer

$$x \u003e 2$$

Question 3

A $$\10$$,000$$ investment is compounded continuously at an annual rate of 5$$\%$$. What will the balance be after 8 years? (Use A$$ = $$Pe^{rt}$)

Accepted Answer

$$A = 10,000 \cdot e$$^{0.05 	imes 8}$$

Question 4

Which property allows you to rewrite $$\log_{a}(MN) as$$ $$\log$$_{a}$$M + \log$$_{a}$$N$$?

Accepted Answer

Product property of logarithms

Question 5

Solve for $$t in $$the equation $$15,000 = 10,000 \cdot e$$^{0.06t}$$.

Accepted Answer

$$t = \frac{\ln(1.5)}{0.06}$$

Question 6

Which of the following is equivalent to $$\log_{5} 125$$?

Accepted Answer

$$3$$

Question 7

Rewrite $$\log_{a} \left( \frac{M^3$ N}{P^2$} \right)$$ using logarithm properties.

Accepted Answer

$$3 \log_{a} M + \log_{a} N - 2 \log_{a} P$$

Question 8

Which statement correctly describes the relationship between exponential and logarithmic functions?

Accepted Answer

They are inverses of each other.

Question 9

If $$f(x) = 2^x$$ and $$g(x) = \log$$_{2}$$x$$, which of the following is true?

Accepted Answer

$$f(g(x)) = x$$

Question 10

A pharmaceutical company needs to triple the amount of a drug in a patient's bloodstream. If the drug increases according to $$A = A_0$ e$$^{kt}$$, and k = 0.2$, how long will it take to triple the initial amount?

Accepted Answer

$$t = \frac{\ln(3)}{0.2}$$

Question 11

Which of the following is the vertical asymptote of $$f(x) = \log_{4}(x-5)$$?

Accepted Answer

$$x = 5$$

Question 12

Condense $$2 \log_{7} x + \log_{7} y - \log_{7} z$$ into a single logarithm.

Accepted Answer

$$\log_{7} \left( \frac{x^2$ y}{z} \right)$$

Question 13

Which formula is used to convert $$\log_{a} x to a $$logarithm with base $$10$$?

Accepted Answer

$$\log_{a} x = \frac{\log x}{\log a}$$

Question 14

Which of the following statements is true about the graph of $$y = 3^x$$?

Accepted Answer

The graph has a horizontal asymptote at $$y = 0.$$

Question 15

Evaluate $$\ln e^5$.

Accepted Answer

$$5$$

Question 16

Which of the following is NOT a property of logarithms?

Accepted Answer

$$\log_{a}(M + N) = \log$$_{a}$$M + \log$$_{a}$$N$$

Question 17

A radioactive substance decays according to $$A = A_0$ e$$^{-0.03t}$$. If the initial amount is 100$ grams, how much remains after 20$ years?

Accepted Answer

$$A = 100 \cdot e$$^{-0.03 	imes 20}$$

Question 18

Which of the following is the correct translation for $$y = 2^x + 4$$?

Accepted Answer

The graph of $$y = 2^x$$ shifted up by 4 units.

Question 19

If $$\log_{a} x = y$$, which equation is equivalent?

Accepted Answer

$$a^y = x$$

Exponential and Logarithmic Functions: Study Guide (Sections 4.1–4.4)

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