- Apply the quotient rule: \(\log_w \left(x^2 - 6\right)^4 - \log_w \left(\sqrt[3]{x^2 + 8}\right)\).
- Use the radical-to-exponent rule: \(\log_w \left(x^2 - 6\right)^4 - \log_w \left(x^2 + 8\right)^{1/3}\).
- Apply the power rule to simplify the exponents: \(4 \log_w (x^2 - 6) - \frac{1}{3} \log_w (x^2 + 8)\).
- Identify the equivalent option: \(\boxed{a}\).
Explanation
- Problem Analysis and Objective
The goal is to expand the logarithmic expression \(\log_w \frac{\left(x^2 - 6\right)^4}{\sqrt[3]{x^2 + 8}}\) using standard logarithmic properties. These properties allow us to simplify the expressions involving products, quotients, and powers. We will rely on the quotient rule, the power rule, and the definition of roots as exponents to reach the correct expansion.
- Applying the Quotient Rule
First, we use the quotient rule for logarithms, which states that \(\log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N\). Applying this to our expression gives:
\[\log_w \frac{\left(x^2 - 6\right)^4}{\sqrt[3]{x^2 + 8}} = \log_w \left(x^2 - 6\right)^4 - \log_w \left(\sqrt[3]{x^2 + 8}\right).\]
- Applying the Power Rule to the First Term
Next, we address the first term using the power rule, \(\log_b (M^k) = k \log_b M\). Applying this to \(\log_w \left(x^2 - 6\right)^4\), we get:
\[\log_w \left(x^2 - 6\right)^4 = 4 \log_w (x^2 - 6).\]
- Rewriting the Radical and Applying the Power Rule
For the second term, \(\log_w \left(\sqrt[3]{x^2 + 8}\right)\), we first express the cube root as a fractional exponent using the rule \(\sqrt[n]{A} = A^{1/n}\):
\[\sqrt[3]{x^2 + 8} = (x^2 + 8)^{1/3}.\]
Now, we apply the power rule again:
\[\log_w (x^2 + 8)^{1/3} = \frac{1}{3} \log_w (x^2 + 8).\]
- Conclusion
Combining the results from the previous steps, we substitute them back into our expression:
\[4 \log_w (x^2 - 6) - \frac{1}{3} \log_w (x^2 + 8).\]
Comparing this result with the given options, it matches option a. Thus, the expression \(\log_w \frac{\left(x^2 - 6\right)^4}{\sqrt[3]{x^2 + 8}}\) is equivalent to \(4 \log_w \left(x^2 - 6\right) - \frac{1}{3} \log_w \left(x^2 + 8\right)\). Therefore, the correct option is \(\boxed{a}\).
Examples
Logarithmic expansion is fundamental in fields like signal processing and acoustics. For instance, when dealing with complex sounds or signals that are represented by ratios of power or magnitude (like in decibel calculations), logarithmic properties help break down complicated quotients into simpler, additive components. This allows engineers to analyze different frequency bands or signal components individually rather than dealing with the entire combined signal at once.