Q: A population of bacteria grows continuously at a rate of $$8\%$$ per hour. If the initial population is $$500$$, what is the population after $$10$$ hours? Use $$A(t) = Pe$$^{rt}$$.

$$A = 500e$$^{0.08 \times 10}$$

Question 1

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

Accepted Answer

$$f(x) = 2^x$$

Question 2

Given $$f(x) = 3x + 2$$ and $$g(x) = \frac{x-2}{3}$$, which statement is true?

Accepted Answer

$$f$$ and $$g$$ are inverses because $$f(g(x)) = x$$ and $$g(f(x)) = x$$

Question 3

If $$f(x) = 2^x$$ and $$g(x) = \log_2$ x$$, what is $$f(g(8))$$?

Accepted Answer

$$8$$

Question 4

Which of the following is the correct inverse of $$f(x) = e$$^{2x}$$?

Accepted Answer

$$f^{-1}(x$$) = \frac{1}{2} \ln x$$

Question 5

Given the exponential function $$f(x) = 5 \cdot 3^x$$, which of the following is a correct table of values for $$x = 0, 1, 2$$?

Accepted Answer

$$x=0: 5,\ x=1: 15,\ x=2: 45$$

Question 6

Solve the logarithmic equation: $$\log_2$(x) = 5.$$

Accepted Answer

$$x = 32$$

Question 7

Convert the logarithmic equation $$\log_{10}(x) = 3 to $$its equivalent exponential form.

Accepted Answer

$$x = 10^3$

Question 8

Convert the exponential equation $$4^x = 20 to a $$logarithmic equation.

Accepted Answer

$$x = \log_4$ 20$$

Question 9

Find the value of $$\log_5$ 125$$ using the change of base formula.

Accepted Answer

$$\frac{\ln 125}{\ln 5}$$

Question 10

Express $$\log_3$(9x) as a $$sum using logarithmic properties.

Accepted Answer

$$\log_3$ 9 + \log_3$ x$$

Question 11

Express $$\log_2$ 32 - \log_2$ 4 as a $$single logarithm.

Accepted Answer

$$\log_2$ 8$$

Question 12

Expand $$\log_5$ \left(\frac{a^3$ b}{c^2$}\right)$$ using logarithmic properties.

Accepted Answer

$$3\log_5$ a + \log_5$ b - 2\log_5$ c$$

Question 13

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

Accepted Answer

$$x = 3$$

Question 14

Solve for $$x$$: $$\log_3$(x-1) = 2.$$

Accepted Answer

$$x = 10$$

Question 15

Solve the exponential equation $$5^{2x} = 125$ by taking the log of both sides.

Accepted Answer

$$x = 1.5$$

Question 16

A $$\1000$ investment grows at an annual rate of 6$$\%$$ compounded quarterly. What is the amount after 5$ years? Use A(t) = P$$ \left[1 + \frac{r}{n}\right]^{nt}$.

Accepted Answer

$$A = 1000 \left[1 + \frac{0.06}{4}\right]^{20}$$

Question 17

A population of bacteria grows continuously at a rate of $$8\%$$ per hour. If the initial population is $$500$$, what is the population after $$10$$ hours? Use $$A(t) = Pe$$^{rt}$$.

Accepted Answer

$$A = 500e$$^{0.08 	imes 10}$$

Question 18

Which of the following is equivalent to $$\log_7$ 49$$?

Accepted Answer

$$2$$

Question 19

If $$\log_b 2 = 0.3010$$ and $$\log_b 5 = 0.6990$$, what is $$\log_b 20$$?

Accepted Answer

$$1.0000$$

Question 20

Which of the following best describes the graph of $$y = \log_2$ x$$?

Accepted Answer

It passes through $$(1,0)$$ and increases slowly for large $$x.$$

Exponential and Logarithmic Functions: College Algebra Study Guide

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