Expand the Logarithmic Expression log of (z^2y)/( cube root of x^5)

$\log (\frac{z^{2} y}{\sqrt[3]{x^{5}}})$

Rewrite $\log (\frac{z^{2} y}{\sqrt[3]{x^{5}}})$ as $\log (z^{2} y) - \log (\sqrt[3]{x^{5}})$ .

$\log (z^{2} y) - \log (\sqrt[3]{x^{5}})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{5}}$ as $x^{\frac{5}{3}}$ .

$\log (z^{2} y) - \log (x^{\frac{5}{3}})$

Expand $\log (x^{\frac{5}{3}})$ by moving $\frac{5}{3}$ outside the logarithm.

$\log (z^{2} y) - (\frac{5}{3} \log (x))$

Combine $\frac{5}{3}$ and $\log (x)$ .

$\log (z^{2} y) - \frac{5 \log (x)}{3}$

Rewrite $\log (z^{2} y)$ as $\log (z^{2}) + \log (y)$ .

$\log (z^{2}) + \log (y) - \frac{5 \log (x)}{3}$

Expand $\log (z^{2})$ by moving $2$ outside the logarithm.

$2 \log (z) + \log (y) - \frac{5 \log (x)}{3}$