Expand the Logarithmic Expression log of (x^2y^5)/( square root of z^3)

$\log (\frac{x^{2} y^{5}}{\sqrt{z^{3}}})$

Rewrite $\log (\frac{x^{2} y^{5}}{\sqrt{z^{3}}})$ as $\log (x^{2} y^{5}) - \log (\sqrt{z^{3}})$ .

$\log (x^{2} y^{5}) - \log (\sqrt{z^{3}})$

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

$\log (x^{2} y^{5}) - \log (z^{\frac{3}{2}})$

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

$\log (x^{2} y^{5}) - (\frac{3}{2} \log (z))$

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

$\log (x^{2} y^{5}) - \frac{3 \log (z)}{2}$

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

$\log (x^{2}) + \log (y^{5}) - \frac{3 \log (z)}{2}$

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

$2 \log (x) + \log (y^{5}) - \frac{3 \log (z)}{2}$

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

$2 \log (x) + 5 \log (y) - \frac{3 \log (z)}{2}$