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

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

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

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

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

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

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

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

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

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

Multiply $4$ by $- 1$ .

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

Rewrite $\log (x^{4} z^{2})$ as $\log (x^{4}) + \log (z^{2})$ .

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

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

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

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

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

Simplify each term.

Apply the distributive property.

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

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

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

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

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

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

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

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

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

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

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