Expand the Logarithmic Expression natural log of ( square root of z)/(xy)

$\ln (\frac{\sqrt{z}}{x y})$

Rewrite $\ln (\frac{\sqrt{z}}{x y})$ as $\ln (\sqrt{z}) - \ln (x y)$ .

$\ln (\sqrt{z}) - \ln (x y)$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{z}$ as $z^{\frac{1}{2}}$ .

$\ln (z^{\frac{1}{2}}) - \ln (x y)$

Expand $\ln (z^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \ln (z) - \ln (x y)$

Rewrite $\ln (x y)$ as $\ln (x) + \ln (y)$ .

$\frac{1}{2} \ln (z) - (\ln (x) + \ln (y))$

Simplify each term.

Combine $\frac{1}{2}$ and $\ln (z)$ .

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

Apply the distributive property.

$\frac{\ln (z)}{2} - \ln (x) - \ln (y)$