Expand the Logarithmic Expression log of ( square root of z^5y^2)/(x^4)

$\log (\frac{\sqrt{z^{5}} y^{2}}{x^{4}})$

Rewrite $\log (\frac{\sqrt{z^{5}} y^{2}}{x^{4}})$ as $\log (\sqrt{z^{5}} y^{2}) - \log (x^{4})$ .

$\log (\sqrt{z^{5}} y^{2}) - \log (x^{4})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{z^{5}}$ as $z^{\frac{5}{2}}$ .

$\log (z^{\frac{5}{2}} y^{2}) - \log (x^{4})$

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

$\log (z^{\frac{5}{2}} y^{2}) - (4 \log (x))$

Multiply $4$ by $- 1$ .

$\log (z^{\frac{5}{2}} y^{2}) - 4 \log (x)$

Rewrite $\log (z^{\frac{5}{2}} y^{2})$ as $\log (z^{\frac{5}{2}}) + \log (y^{2})$ .

$\log (z^{\frac{5}{2}}) + \log (y^{2}) - 4 \log (x)$

Expand $\log (z^{\frac{5}{2}})$ by moving $\frac{5}{2}$ outside the logarithm.

$\frac{5}{2} \log (z) + \log (y^{2}) - 4 \log (x)$

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

$\frac{5}{2} \log (z) + 2 \log (y) - 4 \log (x)$

Combine $\frac{5}{2}$ and $\log (z)$ .

$\frac{5 \log (z)}{2} + 2 \log (y) - 4 \log (x)$