Answer
$\ln x$ + $\frac{1}{2}$$($$\ln y$ - $\ln z$$)$
Work Step by Step
$Expand$ $the$ $expression$:
$\ln (x\sqrt{\frac{y}{z}})$
Apply the First Law of Logarithms
$\ln (x\times \sqrt{\frac{y}{z}})$ = $\ln x$ + $\ln\sqrt{\frac{y}{z}}$
Rewrite the square root for $\sqrt{\frac{y}{z}}$
$\ln x$ + $ln (\frac{y}{z})^\frac{1}{2}$
Apply the Third Law of Logarithms for $ln (\frac{y}{z})^\frac{1}{2}$
$ln (\frac{y}{z})^\frac{1}{2}$ = $\frac{1}{2}$$\ln \frac{y}{z}$
Apply the Second Law of Logarithms for $\frac{1}{2}$$\ln \frac{y}{z}$ (Distribute the half)
$\frac{1}{2}$$\ln (\frac{y}{z})$ = $\frac{1}{2}$$\ln y$ - $\frac{1}{2}$$\ln z$
Assemble the expression
$\ln x$ + $\frac{1}{2}$$\ln y$ - $\frac{1}{2}$$\ln z$
Factor out the $\frac{1}{2}$. We do this since one may put together the $\ln x$ and the $\frac{1}{2}$$\ln y$ when we combine it to one expression
$\ln x$ + $\frac{1}{2}$$($$\ln y$ - $\ln z$$)$
