1. ## simplify

(^16)log(sqrt8)= 0.5* (^16)log(8) = more ?

2. What does (^16) represent? The base of the log?

I suspect you're meant to make use of the fact that 16 = 2^4 and 8 = 2^3

3. Please, confirm which (if any) is the initial expresion you would like to simplify:

$\displaystyle x^{16}\times \log\sqrt{8}$

or

$\displaystyle 16^{ \log\sqrt{8}}$

or

$\displaystyle \log_{16}\sqrt{8}$

4. i meant this one $\displaystyle \log_{16}\sqrt{8}$

5. So you're asking can you simplify $\displaystyle \displaystyle \log_{16}{\sqrt{8}} = \frac{1}{2}\log_{16}{8}$ any further?

$\displaystyle \displaystyle \frac{1}{2}\log_{16}{8} = \frac{1}{2}\log_{16}{(2\cdot 4)}$

$\displaystyle \displaystyle = \frac{1}{2}\log_{16}{2} + \frac{1}{2}\log_{16}{4}$

$\displaystyle \displaystyle = \frac{1}{2}\log_{16}{(16^{\frac{1}{4}})} + \frac{1}{2}\log_{16}{(16^{\frac{1}{2}})}$

$\displaystyle \displaystyle = \frac{1}{8}\log_{16}{16} + \frac{1}{4}\log_{16}{16}$

$\displaystyle \displaystyle = \frac{1}{8} + \frac{1}{4}$

$\displaystyle \displaystyle = \frac{3}{8}$.