Use the properties of logarithms to expand the following expression.
Español
\[
\log \left(\frac{\sqrt{z^{5}}}{x^{3} y}\right)
\]

Each logarithm should involve only one variable and should not have any radicals or exponents.

You may assume that all variables are positive.
\[
\log \left(\frac{\sqrt{z^{5}}}{x^{3} y}\right)=\square
\]
$\square \log \square$

Step 1 ：Given the expression \(\log \left(\frac{\sqrt{z^{5}}}{x^{3} y}\right)\)

Step 2 ：First, we can use the property of logarithms that says \(\log(a/b) = \log(a) - \log(b)\) to split the fraction inside the logarithm: \(\log \left(\sqrt{z^{5}}\right) - \log \left(x^{3} y\right)\)

Step 3 ：Next, we can use the property of logarithms that says \(\log(a*b) = \log(a) + \log(b)\) to split the multiplication inside the second logarithm: \(\log \left(\sqrt{z^{5}}\right) - \log \left(x^{3}\right) - \log \left(y\right)\)

Step 4 ：Then, we can use the property of logarithms that says \(\log(a^n) = n*\log(a)\) to remove the exponents and the square root (which is the same as raising to the power of 1/2): \(\frac{5}{2} \log(z) - 3 \log(x) - \log(y)\)

Step 5 ：So, the expanded form of the given expression is \(\boxed{\frac{5}{2} \log(z) - 3 \log(x) - \log(y)}\)