Question

Solve the equation: $|x-1{|}^{{\log }_3{x}^2-2{\log }_{x}9}=(x-1{)}^7$.

Answer 1

Domain of $x=(0,\infty )-\left\{1\right\}$

$|x-1{|}^{{\log }_3{x}^2-2{\log }_{x}9}=(x-1{)}^7$

$\Rightarrow |x-1{|}^{2{\log }_{3}x-2{\log }_{x}9}=(x-1{)}^7$

$\Rightarrow (x-1{)}^{2({\log }_{3}x-{\log }_{x}9)}=(x-1{)}^7$

$\Rightarrow 2({\log }_{3}x-{\log }_{x}9)=7$

$\Rightarrow 2({\log }_{3}x-2{\log }_{x}3)=7$

We know, ${\log }_{a}x=\frac{1}{{\log }_{x}a}$

$\Rightarrow 2({\log }_{3}x-\frac{2}{{\log }_{3}x})=7$

Let ${\log }_{3}x=a$

$\Rightarrow 2(a-\frac{2}{a})=7$

$\Rightarrow 2a^2-7a-4=0$

$\Rightarrow (2a+1)(a-4)=0$

$a=-\frac{1}{2},a=4$

$\Rightarrow {\log }_{3}x=-\frac{1}{2},{\log }_{3}x=4$

$\Rightarrow x=3^{-1/2},x=3^4$

$\Rightarrow x=\frac{1}{\sqrt{3}},81$