Q: A population of bacteria doubles every $$3$$ hours. If there are $$100$$ bacteria now, how many will there be after $$12$$ hours?

$$100 \times 2$$^{4}$$ = 1600$$

Question 1

Which of the following relations is a function?

Accepted Answer

$$\{(1,2), (2,3), (3,4), (4,5)\}$$

Question 2

Given $$f(x) = \dfrac{3x-1}{5}$$, what is the inverse function $$f^{-1}(x$$)$$?

Accepted Answer

$$f^{-1}(x$$) = \dfrac{5x+1}{3}$$

Question 3

Which of the following functions is one-to-one?

Accepted Answer

$$f(x) = 2x + 1$$

Question 4

A $$5000$$ investment earns $$12\%$$ interest, compounded quarterly. What is the amount after $$6$$ years? Use $$A = P(1 + \frac{r}{n}$$)^{nt}$$.

Accepted Answer

$$A = 5000(1 + \frac{0.12}{4}$$)^{24}$$ \approx 10067$$

Question 5

Find the domain and vertical asymptote of $$f(x) = \log(x+4).$$

Accepted Answer

Domain: $$(-4, \infty)$$, vertical asymptote: $$x = -4$$

Question 6

Express $$\log 2 + \log 3 - \log 6 as a $$single logarithm.

Accepted Answer

$$\log(2 	imes 3 \div 6)$$

Question 7

Solve for $$x$$: $$2^{x+1} = 16$

Accepted Answer

$$x = 3$$

Question 8

Convert $$\log_2$ 7 = x to $$exponential form.

Accepted Answer

$$2^x = 7$$

Question 9

A population of rabbits grows at a rate of $$8\%$$ per day. If there are $$10$$ rabbits now, how many will there be after $$5$$ days? Use $$P(t) = P_0$(1 + r)^t.$$

Accepted Answer

$$P(5) = 10(1.08)^5$ \approx 14.7$$

Question 10

Solve for $$x$$: $$\ln(x) = 2$$

Accepted Answer

$$x = e^2$

Question 11

If $$f(x) = x^2$ + 2$$ and $$g(x) = x - 3$$, what is $$f(g(x))$$?

Accepted Answer

$$f(g(x)) = (x-3)^2$ + 2$$

Question 12

A certain radioactive substance decays at a rate of $$5\%$$ per hour. If the initial amount is $$200$$ grams, how much remains after $$10$$ hours? Use $$A = A_0$ e$$^{kt}$$, k$$ = \ln(0.95)$$.

Accepted Answer

$$A = 200e$$^{10\ln(0.95)}$$ \approx 119.2$$

Question 13

Find the solution to $$\log_3$(x) = 4.$$

Accepted Answer

$$x = 81$$

Question 14

Which of the following is the correct change-of-base formula for $$\log_b a$$?

Accepted Answer

$$\log_b a = \dfrac{\log_c a}{\log_c b}$$

Question 15

If $$f(x) = x^2$ + 2$$ and $$g(x) = x - 3$$, what is $$h(x) = \sqrt{f(g(x))} - 3$$?

Accepted Answer

$$h(x) = \sqrt{(x-3)^2$ + 2} - 3$$

Question 16

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

Accepted Answer

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

Question 17

A population of bacteria doubles every $$3$$ hours. If there are $$100$$ bacteria now, how many will there be after $$12$$ hours?

Accepted Answer

$$100 	imes 2$$^{4}$$ = 1600$$

Question 18

Which of the following is the correct inverse of $$f(x) = 2x + 5$$?

Accepted Answer

$$f^{-1}(x$$) = \dfrac{x-5}{2}$$

Question 19

If $$\log_{10} x = 3$$, what is $$x$$?

Accepted Answer

$$x = 1000$$

Question 20

A certain fish population is modeled by $$P = 1000 \cdot 1.08^t. $$How long will it take for the population to double?

Accepted Answer

$$t = \dfrac{\ln 2}{\ln 1.08} \approx 9.01$$

College Algebra Exam 3 Review: Inverses, Exponential and Logarithmic Functions, and Applications

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