Question

If $\log \frac{x+y}{2}=\frac{1}{2}(\log x+\log y)$, prove that $x=y$

Answer 1

$\log \frac{x+y}{2}=\frac{1}{2}(\log x+\log y)$

$\Rightarrow \log \frac{x+y}{2}=\frac{1}{2}\log xy$

$\Rightarrow \log \left(\frac{x+y}{2}\right)=\log \sqrt{xy}$

$\Rightarrow \frac{x+y}{2}=\sqrt{xy}$

$\Rightarrow x^2+y^2+2xy=4xy$

$\Rightarrow (x-y{)}^2=0$

$\Rightarrow x-y=0$

$\Rightarrow x=y$