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 $a, b > 0$, 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

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

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

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

Solving Logarithmic Equations in College Algebra

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