$\begin{align}  & {{x}^{2}}+{{y}^{2}}=2xy \\ & {{(x+y)}^{2}}-2xy=2xy\left| \text{Note: }{{x}^{2}}+{{y}^{2}}=(x+y)^2-2xy \right. \\ & {{(x+y)}^{2}}=4xy \\ & \text{Take the logarithm of both sides} \\ & \log {{(x+y)}^{2}}=\log 4xy \\ & 2\log (x+y)=\log 4+\log x+\log y \\ & \log (x+y)=\frac{\log 4+\log x+\log y}{2} \\ & \log (x+y)=\frac{\log {{2}^{2}}+\log x+\log y}{2} \\ & \log (x+y)=\frac{2\log 2+\log x+\log y}{2}=\frac{2\log 2}{2}+\frac{\log x}{2}+\frac{\log y}{2} \\ & \log (x+y)=\log 2+\tfrac{1}{2}\log x+\tfrac{1}{2}\log y \\\end{align}$