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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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