just use the power and product rules of logs
$\log_{10}(x^a)=a \log_{10}(x)$
$\log_{10}(a b) = \log_{10}(a)+\log_{10}(b)$
remember
$\sqrt{x}=x^\frac{1}{2}$
$\sqrt[3]{x}=x^\frac{1}{3}$
@nycmath, This may be the last reply I shall ever give you.
Frankly I am fed-up with your posting a raw question, but with your showing no effort whatsoever.
$\log \left[ {\sqrt[3]{{x\sqrt {1 + {y^2}} }}} \right] = \frac{1}{3}\left[ {\log (x) + \frac{1}{2}\log \left( {1 + {y^2}} \right)} \right]$