Answer
$log(x^{\frac{1}{2}}y^{\frac{1}{2}})$
Work Step by Step
Based on the product rule of logarithms, we know that $log_{b}(MN)=log_{b}M+log_{b}N$ (for $M\gt0$ and $N\gt0$).
Therefore, $\frac{1}{2}(log(x)+log(y))=\frac{1}{2}log(xy)$.
According to the power rule of logarithms, we know that $log_{b}M^{p}=plog_{b}M$ (when $b$ and $M$ are positive real numbers, $b\ne1$, and $p$ is any real number).
Therefore, $ \frac{1}{2}log(xy)=log(x^{\frac{1}{2}}y^{\frac{1}{2}})$.
In this case, the given logarithm is a common logarithm with an understood base of 10.