To solve the problem of finding an equivalent expression for the logarithmic expression [tex]\(\log_w \frac{(x^2-6)^4}{\sqrt[3]{x^2+8}}\)[/tex], follow these steps:
1. Use the logarithmic property for division inside a logarithm:
[tex]\[
\log_b \left(\frac{a}{c}\right) = \log_b(a) - \log_b(c)
\][/tex]
Applying this to our expression:
[tex]\[
\log_w \frac{(x^2-6)^4}{\sqrt[3]{x^2+8}} = \log_w \left((x^2 - 6)^4\right) - \log_w \left(\sqrt[3]{x^2 + 8}\right)
\][/tex]
2. Use the logarithmic property for exponents:
[tex]\[
\log_b(a^n) = n \log_b(a)
\][/tex]
Applying this to our expression:
[tex]\[
\log_w \left((x^2 - 6)^4\right) = 4 \log_w (x^2 - 6)
\][/tex]
and
[tex]\[
\log_w \left(\sqrt[3]{x^2 + 8}\right) = \log_w \left((x^2 + 8)^{1/3}\right) = \frac{1}{3} \log_w (x^2 + 8)
\][/tex]
3. Combine the results from steps 1 and 2:
[tex]\[
\log_w \frac{(x^2-6)^4}{\sqrt[3]{x^2+8}} = 4 \log_w (x^2 - 6) - \frac{1}{3} \log_w(x^2 + 8)
\][/tex]
Thus, the equivalent expression is:
[tex]\[
4 \log_w (x^2 - 6) - \frac{1}{3} \log_w (x^2 + 8)
\][/tex]
Next, let's match this result with one of the provided options:
- Option 1:
[tex]\[
4 \log_w \frac{x^2}{1296} - \frac{1}{3} \log_w (2x + 8)
\][/tex]
This does not match our derived expression.
- Option 2:
[tex]\[
4 \log_w (x^2 - 6) - 3 \log_w (x^2 + 8)
\][/tex]
This does not match our derived expression.
- Option 3:
[tex]\[
4 \log_w (x^2 - 6) - \frac{1}{3} \log_w (x^2 + 8)
\][/tex]
This matches our derived expression exactly.
- Option 4:
[tex]\[
4 (\log_w x^2 - \frac{1}{3} \log_w (x^2 + 8) - 6)
\][/tex]
This does not match our derived expression.
Therefore, the expression that is equivalent to [tex]\(\log_w \frac{(x^2-6)^4}{\sqrt[3]{x^2+8}}\)[/tex] is:
[tex]\[
4 \log_w (x^2 - 6) - \frac{1}{3} \log_w (x^2 + 8)
\][/tex]
And the correct option number is 3.
