To determine which expression is equivalent to [tex]\(\log_8 4a\left(\frac{b-4}{c^4}\right)\)[/tex], we need to use the properties of logarithms to simplify the given expression.
1. Apply the logarithm of a product:
[tex]\[
\log_8(4a \cdot \frac{b-4}{c^4}) = \log_8 4a + \log_8 \left(\frac{b-4}{c^4}\right)
\][/tex]
2. Apply the logarithm of a product again:
[tex]\[
\log_8 4a = \log_8 4 + \log_8 a
\][/tex]
Combine these two results:
[tex]\[
\log_8(4a \cdot \frac{b-4}{c^4}) = \log_8 4 + \log_8 a + \log_8 \left(\frac{b-4}{c^4}\right)
\][/tex]
3. Apply the logarithm of a quotient:
[tex]\[
\log_8 \left(\frac{b-4}{c^4}\right) = \log_8 (b-4) - \log_8 (c^4)
\][/tex]
4. Apply the logarithm of a power:
[tex]\[
\log_8 (c^4) = 4 \log_8 c
\][/tex]
Combine these results:
[tex]\[
\log_8 \left(\frac{b-4}{c^4}\right) = \log_8 (b-4) - 4 \log_8 c
\][/tex]
Therefore, plug these results back into our combined expression:
[tex]\[
\log_8(4a \cdot \frac{b-4}{c^4}) = \log_8 4 + \log_8 a + \log_8 (b-4) - 4 \log_8 c
\][/tex]
5. Combine and simplify:
[tex]\[
\log_8(4a \cdot \frac{b-4}{c^4}) = \log_8 4 + \log_8 a - \log_8 (b-4) - 4 \log_8 c
\][/tex]
Therefore, the expression equivalent to [tex]\(\log_8 4a\left(\frac{b-4}{c^4}\right)\)[/tex] is:
[tex]\[
\boxed{\log_8 4+\log_8 a-\log_8 (b-4)-4 \log_8 c}
\][/tex]
