Question

The set of real values of $x$ satisfying the equation ${|x-1|}^{{\log }_3\left(x^2\right)-2{\log }_x\left(9\right)}={(x-1)}^7$ is

• A. $2$
• B. $27,81$
• C. $81$
• D. $2,81$

Answer 1

The correct option is D $2,81$

${|x-1|}^{{\log }_3{x}^2-2{\log }_{x}9}={(x-1)}^7$
L.H.S of the above equation exist when $x\ne 1$
and $x$ cannot be less than $1$
Taking logarithm on both sides, we get
$({\log }_3{x}^2-2{\log }_{x}9)\log (x-1)=7\log (x-1)[\because \log a^m=m\log a]$

$\Rightarrow (2{\log }_{3}x-\frac{4}{{\log }_{3}x})\log (x-1)=7\log (x-1)[\because {\log }_{a}b=\frac{1}{{\log }_{b}a}]$
$\Rightarrow (2{\log }_{3}x-\frac{4}{{\log }_{3}x})\log (x-1)=7\log (x-1)$

$\Rightarrow \log (x-1)=0$ $\Rightarrow x=2$
And $2{\left({\log }_{3}x\right)}^2-7{\log }_{3}x-4=0$
$\Rightarrow {\log }_{3}x=4,\frac{-1}{2}$
$\Rightarrow x=81,\frac{1}{\sqrt{3}}$
Since, $x>1$
Therefore, $x=2,81$
Ans: D