Expand the Logarithmic Expression log of (y^2)/( cube root of xz^4)

$\log (\frac{y^{2}}{\sqrt[3]{x} z^{4}})$

Rewrite $\log (\frac{y^{2}}{\sqrt[3]{x} z^{4}})$ as $\log (y^{2}) - \log (\sqrt[3]{x} z^{4})$ .

$\log (y^{2}) - \log (\sqrt[3]{x} z^{4})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$\log (y^{2}) - \log (x^{\frac{1}{3}} z^{4})$

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

$2 \log (y) - \log (x^{\frac{1}{3}} z^{4})$

Rewrite $\log (x^{\frac{1}{3}} z^{4})$ as $\log (x^{\frac{1}{3}}) + \log (z^{4})$ .

$2 \log (y) - (\log (x^{\frac{1}{3}}) + \log (z^{4}))$

Expand $\log (x^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$2 \log (y) - (\frac{1}{3} \log (x) + \log (z^{4}))$

Expand $\log (z^{4})$ by moving $4$ outside the logarithm.

$2 \log (y) - (\frac{1}{3} \log (x) + 4 \log (z))$

Simplify each term.

Combine $\frac{1}{3}$ and $\log (x)$ .

$2 \log (y) - (\frac{\log (x)}{3} + 4 \log (z))$

Apply the distributive property.

$2 \log (y) - \frac{\log (x)}{3} - (4 \log (z))$

Multiply $4$ by $- 1$ .

$2 \log (y) - \frac{\log (x)}{3} - 4 \log (z)$