Question 1

Solve for $$x in $$the equation:

$$\log_2$ (3x - 1) + \log_2$ (x + 1) = 1$$

Express your answer in exact form.

Accepted Answer

$$x = 1 or$$ $$x = -\dfrac{1}{3} ($$but only $$x = 1 is $$valid)$

Question 2

Which property of logarithms allows you to combine $$\log_b M + \log_b N$$ into a single logarithm?

Accepted Answer

$$\log_b (MN)$$

Question 3

Given $$\log_2$ (3x - 1) + \log_2$ (x + 1) = 1$$, which of the following is the correct first step to combine the logarithms?

Accepted Answer

$$\log_2$ ((3x - 1)(x + 1)) = 1$$

Question 4

After combining the logarithms in $$\log_2$ (3x - 1) + \log_2$ (x + 1) = 1$$, what equation do you get?

Accepted Answer

$$\log_2$ ((3x - 1)(x + 1)) = 1$$

Question 5

What is the equivalent exponential form of $$\log_2$ ((3x - 1)(x + 1)) = 1$$?

Accepted Answer

$$(3x - 1)(x + 1) = 2$$

Question 6

Solve $$(3x - 1)(x + 1) = 2$$ for $$x.$$

Accepted Answer

$$x = 1 or$$ $$x = -\dfrac{1}{3}$$

Question 7

Which of the following values for $$x is an $$extraneous solution for $$\log_2$ (3x - 1) + \log_2$ (x + 1) = 1$$?

Accepted Answer

$$x = -\dfrac{1}{3}$$

Question 8

Why is $$x = -\dfrac{1}{3}$$ not a valid solution for $$\log_2$ (3x - 1) + \log_2$ (x + 1) = 1$$?

Accepted Answer

Because $$3x - 1 = -2 is $$not in the domain of the logarithm

Question 9

If $$\log_2$ (a) + \log_2$ (b) = 1$$, and $$a, b \u003e 0$$, what is the value of $$ab$$?

Accepted Answer

$$ab = 2$$

Question 10

A scientist is measuring the intensity of a sound in decibels, given by $$I = 10 \log_{10} \left(\dfrac{P}{P_0$}\right). If $$the power $$P is $$doubled, by how many decibels does the intensity increase?

Accepted Answer

$$10 \log_{10} 2$$

Question 11

Which of the following is NOT a valid property of logarithms?

Accepted Answer

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

Question 12

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

Accepted Answer

$$x = 32$$

Question 13

Which of the following is the correct domain for the function $$f(x) = \log_2$ (3x - 1)$$?

Accepted Answer

$$x \u003e \dfrac{1}{3}$$

Question 14

If $$\log_2$ (3x - 1) + \log_2$ (x + 1) = 1$$, what is the restriction on $$x$$ for the equation to be defined?

Accepted Answer

$$x \u003e -1$$ and $$x \u003e \dfrac{1}{3}$$

Solving Logarithmic Equations in College Algebra

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