Question

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

Answer 1

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