Expanding a logarithmic expression: Problem type 2

Use the properties of logarithms to expand the following expression.
\[
\log \left(\frac{x^{7}}{z \sqrt[3]{y^{2}}}\right)
\]

Each logarithm should involve only one variable and should not have any radicals or exponents. You may assume that all variables are positive.
\[
\log \left(\frac{x^{7}}{z \sqrt[3]{y^{2}}}\right)=\square
\]

Step 1 ：The given expression is a logarithm of a fraction, which can be split into the difference of two logarithms.

Step 2 ：The numerator of the fraction is \(x^7\), which can be written as \(7*\log(x)\) using the property of logarithms that \(\log(a^b) = b*\log(a)\).

Step 3 ：The denominator of the fraction is \(z*(y^{2/3})\), which can be split into \(\log(z) + (2/3)*\log(y)\) using the same property of logarithms.

Step 4 ：Therefore, the expanded form of the given expression is \(7*\log(x) - \log(z) - (2/3)*\log(y)\).

Step 5 ：\(\boxed{\log \left(\frac{x^{7}}{z \sqrt[3]{y^{2}}}\right) = 7\log(x) - \log(z) - \frac{2}{3}\log(y)}\)