Question

Use the information to answer the question.

Information
Britta solves the logarithmic equation Log base 2 of left-parenthesis x squared minus 16 right-parenthesis EndLog equals 1. She wants to check her work by plugging the solution back into the equation to determine if her solution is valid.

Question
Which statement explains why x equals 0 is not a valid solution?

Answer options with 4 options
A.
0 is not a multiple of 2.

B.
The value of x cannot be 0.

C.
The range of a logarithm does not include 1.

D.
The domain of a logarithm does not include negative 16.

Answer 1

To determine why \( x = 0 \) is not a valid solution for the given logarithmic equation \( \log_2(x^2 - 16) = 1 \), we need to analyze the expression inside the logarithm.

First, we need to find the values of \( x \) that make the argument of the logarithm, \( x^2 - 16 \), positive because the logarithm is only defined for positive arguments.

1. Set up the inequality:
\[
x^2 - 16 > 0
\]
2. Solve the inequality:
\[
x^2 > 16
\]
Taking the square root of both sides gives:
\[
x > 4 \quad \text{or} \quad x < -4.
\]

Since \( x = 0 \) does not satisfy either of these conditions (i.e., 0 is neither greater than 4 nor less than -4), it results in a negative argument for the logarithm, making it undefined.

Now, analyzing the provided answer options:
- A. 0 is not a multiple of 2. (Not relevant to the argument of the logarithm)
- B. The value of x cannot be 0. (This statement is correct in the context that it leads to a negative argument.)
- C. The range of a logarithm does not include 1. (This is not correct; the logarithm can equal 1.)
- D. The domain of a logarithm does not include negative 16. (This doesn't explain why \( x = 0 \) is invalid.)

Thus, the best answer is:

**B. The value of x cannot be 0.**