Sure, let's expand the given logarithmic expression step by step using properties of logarithms.
Given expression:
[tex]\[
\log \left(\sqrt[3]{\frac{(x+7)^4}{x^5}}\right)
\][/tex]
1. Apply the radical rule of logarithms:
The cube root of an expression can be written as an exponent of [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
\sqrt[3]{\frac{(x+7)^4}{x^5}} = \left(\frac{(x+7)^4}{x^5}\right)^{\frac{1}{3}}
\][/tex]
Thus, the logarithmic expression becomes:
[tex]\[
\log \left(\left(\frac{(x+7)^4}{x^5}\right)^{\frac{1}{3}}\right)
\][/tex]
2. Apply the power rule of logarithms:
[tex]\(\log(a^b) = b \log(a)\)[/tex]. So, we can bring the [tex]\(\frac{1}{3}\)[/tex] exponent to the front of the logarithm:
[tex]\[
\log \left(\left(\frac{(x+7)^4}{x^5}\right)^{\frac{1}{3}}\right) = \frac{1}{3} \log \left(\frac{(x+7)^4}{x^5}\right)
\][/tex]
3. Apply the quotient rule of logarithms:
[tex]\(\log\left(\frac{a}{b}\right) = \log(a) - \log(b)\)[/tex]. Therefore, we can split the logarithm of the fraction:
[tex]\[
\frac{1}{3} \log \left(\frac{(x+7)^4}{x^5}\right) = \frac{1}{3} \left( \log((x+7)^4) - \log(x^5) \right)
\][/tex]
4. Apply the power rule of logarithms again:
[tex]\(\log(a^b) = b \log(a)\)[/tex]. We can move the exponents in [tex]\(\log((x+7)^4)\)[/tex] and [tex]\(\log(x^5)\)[/tex] to the front:
[tex]\[
\frac{1}{3} \left( \log((x+7)^4) - \log(x^5) \right) = \frac{1}{3} \left( 4 \log(x+7) - 5 \log(x) \right)
\][/tex]
5. Distribute the [tex]\(\frac{1}{3}\)[/tex] across the terms within the parentheses:
[tex]\[
\frac{1}{3} \left( 4 \log(x+7) - 5 \log(x) \right) = \frac{4}{3} \log(x+7) - \frac{5}{3} \log(x)
\][/tex]
Therefore, the expanded form of the given logarithmic expression is:
[tex]\[
\boxed{\frac{4}{3} \log(x+7) - \frac{5}{3} \log(x)}
\][/tex]
