Answer
The equivalent expression for $\log \sqrt[4]{\frac{{{z}^{2}}}{{{x}^{3}}y}}$is $\frac{1}{4}\left( 2\log z-3\log x-\log y \right)$.
Work Step by Step
$\log \sqrt[4]{\frac{{{z}^{2}}}{{{x}^{3}}y}}$
Evaluate the expression in terms of a sum as follows.
$\log \sqrt[4]{\frac{{{z}^{2}}}{{{x}^{3}}y}}=\log {{\left( \frac{{{z}^{2}}}{{{x}^{3}}y} \right)}^{\frac{1}{4}}}$
Apply the power rule for logarithms as follows.
$\log \sqrt[4]{\frac{{{z}^{2}}}{{{x}^{3}}y}}=\frac{1}{4}\log \frac{{{z}^{2}}}{{{x}^{3}}y}$
Apply the quotient rule for logarithms as follows.
$\begin{align}
& \log \sqrt[4]{\frac{{{z}^{2}}}{{{x}^{3}}y}}=\frac{1}{4}\left( \log {{z}^{2}}-\log {{x}^{3}}y \right) \\
& =\frac{1}{4}\left( \log {{z}^{2}}-\left( \log {{x}^{3}}+\log y \right) \right)
\end{align}$
Again, apply power rule and simplify,
$\log \sqrt[4]{\frac{{{z}^{2}}}{{{x}^{3}}y}}=\frac{1}{4}\left( 2\log z-3\log x-\log y \right)$
