Question

The value of $x$ satisfying the equation ${|x-1|}^{{\log }_3{x}^2-2{\log }_{9}x}=(x-1{)}^7$ is

• A. $3^4$
• B. $3^5$
• C. $3^6$
• D. $3^7$

Answer 1

The correct option is D $3^7$

$\left(i\right){\log }_{a}b$ hold good if $a>0,a\ne 1,a>1$

i.e. $a-1>0$

$\therefore |a-1|=a-1$

$\left(ii\right)a^b>0\forall b\in R$

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

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

$\Rightarrow 2{\log }_{3}x-2{\log }^{x}9=7$ taking log at base $(x-1)$ bothsides. $lo{g}_{x}3=7,x=3^7$