- We use the logarithmic product rule to separate \(\log_8(4a \cdot \frac{b-4}{c^4})\) into \(\log_8(4a) + \log_8(\frac{b-4}{c^4})\).
- We further expand \(\log_8(4a)\) to \(\log_8 4 + \log_8 a\).
- We apply the quotient rule to \(\log_8(\frac{b-4}{c^4})\) to get \(\log_8(b-4) - \log_8(c^4)\).
- Using the power rule, \(\log_8(c^4)\) becomes \(4\log_8 c\), resulting in the final expression: \(\boxed{\log _8 4 + \log _8 a + (\log _8(b-4) - 4 \log _8 c)}\).
Explanation
- Expanding the Logarithmic Expression
We are asked to find the expression equivalent to \(\log _8 \left( 4 a \left( \frac{b-4}{c^4} \right) \right)\). To do this, we will use the standard properties of logarithms:
- Product Rule: \(\log_b(MN) = \log_b M + \log_b N\)
- Quotient Rule: \(\log_b(M/N) = \log_b M - \log_b N\)
- Power Rule: \(\log_b(M^k) = k \log_b M\)
Starting with the expression:
\[\log _8 \left( 4 a \left( \frac{b-4}{c^4} \right) \right)\]
We can treat \(4a\) as \(M\) and \(\frac{b-4}{c^4}\) as \(N\) in the product rule, giving:
\[\log_8(4a) + \log_8\left(\frac{b-4}{c^4}\right)\]
Now, expand \(\log_8(4a)\) using the product rule again:
\[\log_8 4 + \log_8 a\]
Next, expand the second term \(\log_8\left(\frac{b-4}{c^4}\right)\) using the quotient rule:
\[\log_8(b-4) - \log_8(c^4)\]
Finally, apply the power rule to \(\log_8(c^4)\):
\[\log_8(c^4) = 4 \log_8 c\]
Putting all these together, we get:
\[(\log_8 4 + \log_8 a) + (\log_8(b-4) - 4 \log_8 c) = \log_8 4 + \log_8 a + (\log_8(b-4) - 4 \log_8 c)\]
Comparing this result to the given options, we find it matches the second option.
Examples
Logarithmic properties are fundamental in many fields, such as chemistry and finance. For instance, in chemistry, pH calculations involve logarithms: \(pH = -\log[H^+]\). When dealing with complex formulas in titration or chemical equilibrium involving multiple species, we often use these expansion rules to simplify expressions and isolate individual concentration variables. Mastering these properties allows scientists to perform rapid mental calculations and simplify complex models efficiently.