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

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

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

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{z^{2}}$ as $z^{\frac{2}{3}}$ .

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

Rewrite $\log (z^{\frac{2}{3}} y^{2})$ as $\log (z^{\frac{2}{3}}) + \log (y^{2})$ .

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

Expand $\log (z^{\frac{2}{3}})$ by moving $\frac{2}{3}$ outside the logarithm.

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

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

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

Combine $\frac{2}{3}$ and $\log (z)$ .

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