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

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

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

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

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

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

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

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

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

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

Multiply $4$ by $- 1$ .

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

Rewrite $\log (x^{\frac{1}{2}} y^{\frac{2}{3}})$ as $\log (x^{\frac{1}{2}}) + \log (y^{\frac{2}{3}})$ .

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

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

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

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

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

Simplify each term.

Combine $\frac{1}{2}$ and $\log (x)$ .

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

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

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