Question

If log $\left(\frac{x-y}{5}\right)=\frac{1}{2}$ log $x+\frac{1}{2}$ log $y$,
Show that $x^2+y^2=27xy$.

Answer 1

$log\left(\frac{x-y}{5}\right)=\frac{1}{2}logx+\frac{1}{2}logy$

$\Rightarrow log\left(\frac{x-y}{5}\right)=log\sqrt{x}+log\sqrt{y}=log\left(\sqrt{xy}\right)$

$\Rightarrow \left(\frac{x-y}{5}\right)=\sqrt{xy}\Rightarrow \frac{{(x-y)}^2}{25}=xy$

$\Rightarrow x^2+y^2-2xy=25xy$

$\Rightarrow x^2+y^2=27xy$

Proved.