To expand the expression [tex]\(\log \left(\frac{\sqrt[5]{x}}{y}\right)\)[/tex], we will use the properties of logarithms. Specifically, we will use the quotient rule and the power rule.
### Steps:
1. Quotient Rule:
The logarithm of a quotient is the difference of the logarithms.
[tex]\[
\log \left(\frac{a}{b}\right) = \log a - \log b
\][/tex]
Applying this to [tex]\(\log \left(\frac{\sqrt[5]{x}}{y}\right)\)[/tex]:
[tex]\[
\log \left(\frac{\sqrt[5]{x}}{y}\right) = \log (\sqrt[5]{x}) - \log (y)
\][/tex]
2. Power Rule:
The logarithm of a power is the exponent times the logarithm of the base.
[tex]\[
\log (a^b) = b \log a
\][/tex]
Applying this to [tex]\(\log (\sqrt[5]{x})\)[/tex]:
[tex]\(\sqrt[5]{x}\)[/tex] can be written as [tex]\(x^{1/5}\)[/tex]. Therefore:
[tex]\[
\log (\sqrt[5]{x}) = \log (x^{1/5}) = \frac{1}{5} \log (x)
\][/tex]
3. Combine Results:
Substitute this result back into the first expression we got:
[tex]\[
\log \left(\frac{\sqrt[5]{x}}{y}\right) = \frac{1}{5} \log (x) - \log (y)
\][/tex]
### Conclusion:
The expanded form of [tex]\(\log \left(\frac{\sqrt[5]{x}}{y}\right)\)[/tex] is:
[tex]\[
\boxed{\frac{1}{5} \log x - \log y}
\][/tex]
Thus, the correct option from the given choices is:
[tex]\[
\frac{1}{5} \log x - \log y
\][/tex]
