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

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

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

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

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

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

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

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

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

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

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

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

Simplify each term.

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

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

Apply the distributive property.

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

Multiply $5$ by $- 1$ .

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