Answer
$\displaystyle \frac{1}{2}+\frac{1}{2}\ln x $
Work Step by Step
$\ln\sqrt{ex}=\quad $... apply: $\sqrt{ab}=(ab)^{1/2}$
$=\ln(ex)^{1/2}\quad $...apply the Power Rule: $\quad \log_{b}(M^{p})=p\cdot\log_{b}\mathrm{M}$
$=\displaystyle \frac{1}{2}\ln(ex)\quad $...apply the Product Rule: $\quad \log_{b}(MN)=\log_{b}\mathrm{M}+\log_{b}\mathrm{N}$
$=\displaystyle \frac{1}{2}(\ln e+\ln x)\quad $...apply the basic property: $\log_{b}b=1$
$=\displaystyle \frac{1}{2}(1+\ln x)\quad $... distribute
=$\displaystyle \frac{1}{2}+\frac{1}{2}\ln x $
