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

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

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

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

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

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

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

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

Multiply $5$ by $- 1$ .

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

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

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

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

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

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

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