Q. The value of x satisfying the equation $\left|x-1\right|^{log_{3} \,x^2 -2 log_{x} \,9}=\left(x-1\right)^{7}$ is
Complex Numbers and Quadratic Equations
Solution:
(i) $log_{a} \,b$ hold good if $a >\,0$, $a \ne1$, $a >\,1$
i.e. $a-1 >\,0$
$\therefore |a-1|=a-1$
(ii) $a^{b} >\,0 \, \forall-b \, \in\,R$
Now, from given
$\left|x-1\right|^{log_{3} \,x^2-2log_{x} 9}=\left(x-1\right)^{7} $
$\Rightarrow \left(x-1\right)^{log_{3} x^2-2log_{x} 9}=\left(x-1\right)^{7}$
Taking log at base $(x - 1)$ on both sides
$\Rightarrow \, 2 log_{3} x -2 log_{x}9=7$
$\Rightarrow \, 2log_{3}x-2log_{x} 3^{2}=7$
$\Rightarrow 2t^{2}-7t-4=0$, where $t =log_3{x}$
$\Rightarrow (2t+1)(t-4)=0$
$\therefore 3^{t}=x$
$\Rightarrow t=4 $ or $t=-1/2$
or $x=81, \frac{1}{\sqrt{3}}$, but $x >\,1$ by Def. (i)
$\therefore $ Only possible value of $x=81$
$\therefore $ Only possible value of $x=81$