Question

Solve $2{\log }_{x}a+{\log }_{ax}a+3{\log }_{b}a=0$ where $a>O,b=a^{2}x$

Answer 1

The given equation can be rewritten as
$\frac{2}{{\log }_{a}x}+\frac{1}{{\log }_{a}ax}+\frac{3}{{\log }_a{a}^{2}x}=0$
$[\because b=a^{2}x$, given $]$
$\Rightarrow \frac{2}{{\log }_{a}x}+\frac{1}{1+{\log }_{a}x}+\frac{3}{2+{\log }_{a}x}=0$
$\Rightarrow \frac{2}{t}+\frac{1}{1+t}+\frac{3}{2+t}=0$, where $t={\log }_{a}x$
$\Rightarrow 2(1+t)(2+t)+3t(1+t)+t(2+t)=0$
$\Rightarrow 6t^2+11t+4=0$
$\Rightarrow (2t+1)(3t+4)=0$
$\Rightarrow t=-\frac{1}{2}$ or $t=-\frac{4}{3}$
$\therefore {\log }_{a}x=-\frac{1}{2}$ or ${\log }_{a}x=-\frac{4}{3}$
$\Rightarrow x=a^{-1/2}$ or $x=a^{-4/3}$