Question 1

Which of the following is the correct definition of the logarithmic function for $$x \u003e 0$$, $$a \u003e 0$$, and $$a 
eq 1$$?

Accepted Answer

$$\log_a x = y if $$and only if $$a^y = x$$

Question 2

Which property of logarithms allows you to write $$\log_a (xy) as a $$sum of two logarithms?

Accepted Answer

Product Rule for Logarithms: $$\log_a (xy) = \log_a x + \log_a y$$

Question 3

Given the equation $$2^x = 10$$, which of the following is the correct way to solve for $$x$$ using logarithms?

Accepted Answer

$$x = \log_2$ 10$$

Question 4

What is the value of $$x$$ that solves the equation $$\log_3$ (x - 1) = 2$$?

Accepted Answer

Question 5

Which of the following is the correct application of the change of base formula for $$\log_5$ 20$$?

Accepted Answer

$$\log_5$ 20 = \frac{\log_{10} 20}{\log_{10} 5}$$

Question 6

A biologist is studying bacteria growth modeled by $$N = 500 \cdot 2$$^{t}$$, where N$ is the number of bacteria after t$ hours. How many hours will it take for the population to reach 4000$?

Accepted Answer

$$t = 4$$

Question 7

Which property of logarithms is used to simplify $$\log_a (x^r)$$?

Accepted Answer

Power Rule for Logarithms: $$\log_a (x^r) = r \log_a x$$

Question 8

Solve for $$x$$: $$3^{2x} = 81$.

Accepted Answer

$$x = 2$$

Question 9

If $$\log_2$ x = 5$$, what is the value of $$x$$?

Accepted Answer

$$x = 32$$

Question 10

Which of the following is a correct step when solving $$\log_4$ (x + 1) + \log_4$ (x - 1) = 2$$?

Accepted Answer

Combine using the product rule: $$\log_4$ [(x + 1)(x - 1)] = 2$$

Question 11

A medical researcher models the spread of a virus with $$P = 1000 \cdot 3$$^{t}$$, where P$ is the number of cases after t$ days. How many days will it take for the cases to reach 27$$,000$$?

Accepted Answer

Question 12

Which of the following is the correct way to solve $$5^x = 12$$ for $$x$$?

Accepted Answer

$$x = \log_5$ 12$$

Question 13

If $$\log_7$ (x - 2) = 3$$, what is the value of $$x$$?

Accepted Answer

$$x = 343$$

Question 14

Which of the following is a correct use of the quotient rule for logarithms?

Accepted Answer

$$\log_a (x/y) = \log_a x - \log_a y$$

Question 15

Solve for $$x$$: $$\log_2$ (x^2$ - 1) = 3.$$

Accepted Answer

$$x = \pm 3$$

Question 16

Which of the following is a necessary step when solving logarithmic equations?

Accepted Answer

Check for extraneous solutions by verifying that all arguments of logarithms are positive.

Question 17

A chemical reaction follows $$C = 200 \cdot 0.5^t$$, where $$C is $$the concentration after $$t$$ hours. How long until the concentration drops to $$25$$?

Accepted Answer

$$t = 5$$

Question 18

Which of the following is the correct way to solve $$\log_2$ x + \log_2$ (x - 2) = 3$$?

Accepted Answer

Combine using the product rule: $$\log_2$ [x(x - 2)] = 3$$

Question 19

If $$\log_5$ x = 2$$, what is the value of $$x$$?

Accepted Answer

$$x = 25$$

Exponential and Logarithmic Equations: Definitions, Properties, and Solution Methods

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