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

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

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

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

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

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

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

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

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

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

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

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

Simplify each term.

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

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

Apply the distributive property.

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

Multiply $4$ by $- 1$ .

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