(^16)log(sqrt8)= 0.5* (^16)log(8) = more ?
So you're asking can you simplify $\displaystyle \displaystyle \log_{16}{\sqrt{8}} = \frac{1}{2}\log_{16}{8}$ any further?
$\displaystyle \displaystyle \frac{1}{2}\log_{16}{8} = \frac{1}{2}\log_{16}{(2\cdot 4)}$
$\displaystyle \displaystyle = \frac{1}{2}\log_{16}{2} + \frac{1}{2}\log_{16}{4}$
$\displaystyle \displaystyle = \frac{1}{2}\log_{16}{(16^{\frac{1}{4}})} + \frac{1}{2}\log_{16}{(16^{\frac{1}{2}})}$
$\displaystyle \displaystyle = \frac{1}{8}\log_{16}{16} + \frac{1}{4}\log_{16}{16}$
$\displaystyle \displaystyle = \frac{1}{8} + \frac{1}{4}$
$\displaystyle \displaystyle = \frac{3}{8}$.