Expand the Logarithmic Expression log of cube root of (xy^4)/(z^2)

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{\frac{x y^{4}}{z^{2}}}$ as ${(\frac{x y^{4}}{z^{2}})}^{\frac{1}{3}}$ .

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

Expand $\log ({(\frac{x y^{4}}{z^{2}})}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

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

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

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

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

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

Multiply $2$ by $- 1$ .

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

Rewrite $\log (x y^{4})$ as $\log (x) + \log (y^{4})$ .

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

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

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

Apply the distributive property.

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

Simplify.

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

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

Multiply $\frac{1}{3} (4 \log (y))$ .

Combine $4$ and $\frac{1}{3}$ .

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

Combine $\frac{4}{3}$ and $\log (y)$ .

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

Multiply $\frac{1}{3} (- 2 \log (z))$ .

Combine $- 2$ and $\frac{1}{3}$ .

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

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

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

Move the negative in front of the fraction.

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