2016/029) Solve $2\log_x a + \log_{ax} a + 3\log_b a =0$ where $a > 0$ and $b=a^2 x$

We have $\frac{2}{\log_a x}  + \frac{1}{\log_a ax} + \frac{3}{\log_a a^2 }x= 0$
$=> \frac{2}{\log_a x} + \frac{1}{\log_a x + 1} + \frac{3}{\log_a x + 2}=0$
$=>  \frac{2}{p } + \frac{1}{1+p } + \frac{3}{2 + p}= 0$ where $p = \log_a x$
$=>2(1+p)(2+p) + p(2+p) + 3p(1+p) = 0$
$=> 4 + 6p + 2 p^2 + 2p + p^2 + 3p + 3p^2 = 6p^2 + 11p + 4$
$ = 6p^2 + 8p + 3p+4 = (3p+4) (2p+1)=0$
$=> p = \frac{-4}{3}$ or $\frac{-1}{2}$
Hence $x= a^{-\frac{4}{3}}$ or $a^{\frac{-1}{2}}$