Q. If $x_1, x_2\left(x_2>x_1\right)$ satisfy the equation, $5 \log _{x / 9}(x)+\log _{9 / x}\left(x^3\right)+8 \log _{9 x^2}\left(x^2\right)=2$ then
Complex Numbers and Quadratic Equations
Solution:
$\frac{5 \log _3 x}{\log _3 x-2}+\frac{3 \log _3 x}{2-\log _3 x}+\frac{16 \log _3 x}{2+2 \log _3 x}=2$
$\text { let } \log _3 x = t $
$\frac{5 t }{ t -2}-\frac{3 t }{ t -2}+\frac{8 t }{1+ t }=2 $
$\Rightarrow \frac{5 t -3 t }{ t -2}+\frac{8 t }{1+ t }=2 $
$\Rightarrow \frac{2 t }{ t -2}+\frac{8 t }{1+ t }=2$
$\Rightarrow 4 t^2-6 t+2=0 $
$ \Rightarrow 2 t^2-3 t+1=0$
