Which expression is equivalent to [tex]\log_8 \left(4 a \left(\frac{b-4}{c^4}\right)\right)[/tex]?

A. [tex]\log_8 4 + \log_8 a - \log_8 (b-4) - 4 \log_8 c[/tex]
B. [tex]\log_8 4 + \log_8 a + \log_8 (b-4) - 4 \log_8 c[/tex]
C. [tex]\log_8 4a + \log_8 b - 4 - 4 \log_8 c - 4[/tex]
D. [tex]\log_8 4a - \log_8 (b-4) - \log_8 4c[/tex]



Answer :

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]