If it is known that $x=9$ is a root of the equation 
$lo{g}_x(x^2+15a^2)-lo{g}_x(a-2)=lo{g}_x\left(\frac{8ax}{a-2}\right)$, Find the other root ?

${\log }_x(15a^2+x^2)-lo{g}_x(a-2)={\log }_x\left(\frac{8ax}{a-2}\right)$

${\log }_x\left(\frac{15a^2+x^2}{a-2}\right)={\log }_x\left(\frac{8ax}{a-2}\right)$

$\Rightarrow \frac{15a^2+x^2}{a-2}=\frac{8ax}{a-2}$
$\Rightarrow 15a^2+x^2-8ax=0$    ....(1)
Given $x=9$ is a root 
$\Rightarrow 5a^2-24a+27=0$
$\Rightarrow (5a-9)(a-3)=0$
$\Rightarrow a=3,a=\frac{9}{5}$
From the given eqn it follows that
$a-2>0\Rightarrow a>2$
Hence, $a=3$
Put this value in (1), we get
$x^2-24x+135=0$
$\Rightarrow (x-15)(x-9)=0$
$\Rightarrow x=15,9$
Hence, the other root is 15.