Expand the Logarithmic Expression log of (z^3 cube root of x^5)/(y^2)

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

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

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

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

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

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

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

Multiply $2$ by $- 1$ .

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

Rewrite $\log (z^{3} x^{\frac{5}{3}})$ as $\log (z^{3}) + \log (x^{\frac{5}{3}})$ .

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

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

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

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

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

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

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