1/2 ( x + y ) will equal to ( x + y2 ) if -

Answer: Option C

$$\begin{array}{rl}& \frac{1}{2}\left(\log x+\log y\right)=\log \left(\frac{x+y}{2}\right)\\ & \Rightarrow \frac{1}{2}\log \left(xy\right)=\log \left(\frac{x+y}{2}\right)\\ & \Rightarrow \log {\left(xy\right)}^{\frac{1}{2}}=\log \left(\frac{x+y}{2}\right)\\ & \Rightarrow {\left(xy\right)}^{\frac{1}{2}}=\left(\frac{x+y}{2}\right)\\ & \Rightarrow xy={\left(\frac{x+y}{2}\right)}^2\\ & \Rightarrow 4xy=x^2+y^2+2xy\\ & \Rightarrow x^2+y^2-2xy=0\\ & \Rightarrow {\left(x-y\right)}^2=0\\ & \Rightarrow x-y=0\\ & \Rightarrow x=y\end{array}$$