To solve the logarithmic expression [tex]\(\log_4 \frac{8}{x^2}\)[/tex], we will use properties of logarithms to simplify it step-by-step.
First, we'll apply the logarithmic property dealing with the division inside the logarithm:
[tex]\[
\log_b \left(\frac{m}{n}\right) = \log_b (m) - \log_b (n)
\][/tex]
Applying this property, we have:
[tex]\[
\log_4 \left(\frac{8}{x^2}\right) = \log_4 (8) - \log_4 (x^2)
\][/tex]
Next, we use the property of logarithms that deals with exponents:
[tex]\[
\log_b (m^n) = n \cdot \log_b (m)
\][/tex]
Here, [tex]\( \log_4 (x^2) \)[/tex] can be rewritten using this property:
[tex]\[
\log_4 (x^2) = 2 \cdot \log_4 (x)
\][/tex]
Now substituting back into our expression, we get:
[tex]\[
\log_4 (8) - \log_4 (x^2) = \log_4 (8) - 2 \log_4 (x)
\][/tex]
Thus, the simplified expression for [tex]\(\log_4 \frac{8}{x^2}\)[/tex] is:
[tex]\[
\log_4 (8) - 2 \log_4 (x)
\][/tex]
Comparing this result with the given options:
A. [tex]\(2 \log_4 8 - \log_4 x\)[/tex]
B. [tex]\(\log_4 8 + 2 \log_4 x\)[/tex]
C. [tex]\(2 \log_4 8 + \log_4 x\)[/tex]
D. [tex]\(\log_4 8 - 2 \log_4 x\)[/tex]
We see that option D matches the simplified expression we obtained:
[tex]\[
\log_4 (8) - 2 \log_4 (x)
\][/tex]
So, the correct answer is:
D. [tex]\(\log_4 8 - 2 \log_4 x\)[/tex]
