To find the expression equivalent to [tex]\(\log_8 \left(4a \frac{b-4}{c^4}\right)\)[/tex], we will use the properties of logarithms: the product rule and the quotient rule.
### Step-by-Step Solution:
1. Expression Inside the Logarithm: First, write down the expression inside the logarithm:
[tex]\[
4a \left(\frac{b-4}{c^4}\right)
\][/tex]
2. Applying the Product Rule: The logarithm of a product is the sum of the logarithms of the factors. Hence, we use:
[tex]\[
\log_8 \left(4a \frac{b-4}{c^4}\right) = \log_8 \left(4a\right) + \log_8 \left(\frac{b-4}{c^4}\right)
\][/tex]
3. Separating the Logarithms using Product Rule: Break down [tex]\(\log_8 (4a)\)[/tex] and [tex]\(\log_8 \left(\frac{b-4}{c^4}\right)\)[/tex]:
[tex]\[
\log_8 (4a) + \log_8 \left(\frac{b-4}{c^4}\right) = (\log_8 4 + \log_8 a) + \log_8 \left(\frac{b-4}{c^4}\right)
\][/tex]
4. Applying the Quotient Rule: The logarithm of a quotient is the difference of the logarithms. Thus, we get:
[tex]\[
\log_8 \left(\frac{b-4}{c^4}\right) = \log_8 (b-4) - \log_8 (c^4)
\][/tex]
5. Applying the Power Rule: The logarithm of a power is the exponent times the logarithm of the base. Therefore, [tex]\(\log_8 (c^4)\)[/tex] becomes:
[tex]\[
\log_8 (c^4) = 4 \log_8 c
\][/tex]
6. Combining All Parts: Substituting the results back, we get:
[tex]\[
\log_8 (4a) + \log_8 \left(\frac{b-4}{c^4}\right) = (\log_8 4 + \log_8 a) + (\log_8 (b-4) - 4 \log_8 c)
\][/tex]
7. Final Expression: Combine all these parts together:
[tex]\[
\log_8 4 + \log_8 a + \log_8 (b-4) - 4 \log_8 c
\][/tex]
Hence, the expression equivalent to [tex]\(\log_8 \left(4a \frac{b-4}{c^4}\right)\)[/tex] is:
[tex]\[
\log_8 4 + \log_8 a + \log_8 (b-4) - 4 \log_8 c
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{\log_8 4+\log_8 a+\left(\log_8(b-4)-4 \log_8 c\right)}
\][/tex]
