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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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