Expand the Logarithmic Expression log of x/( cube root of yz^4)

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

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

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

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

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

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

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

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

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

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

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

Simplify each term.

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

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

Apply the distributive property.

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

Multiply $4$ by $- 1$ .

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