Question

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

• A. $27$
• B. $81$
• C. $9$
• D. $3$

Answer 1

Answer: (B)

(i) $lo{g}_{a}b$ hold good if $a>0$, $a\ne 1$, $a>1$
i.e. $a-1>0$
$\therefore |a-1|=a-1$
(ii) $a^b>0\forall -b\in R$
Now, from given
${\left|x-1\right|}^{lo{g}_3{x}^2-2lo{g}_{x}9}={\left(x-1\right)}^7$
$\Rightarrow {\left(x-1\right)}^{lo{g}_3{x}^2-2lo{g}_{x}9}={\left(x-1\right)}^7$
Taking log at base $(x-1)$ on both sides
$\Rightarrow 2lo{g}_{3}x-2lo{g}_{x}9=7$
$\Rightarrow 2lo{g}_{3}x-2lo{g}_x{3}^2=7$
$\Rightarrow 2t^2-7t-4=0$, where $t=lo{g}_{3}x$
$\Rightarrow (2t+1)(t-4)=0$
$\therefore 3^t=x$
$\Rightarrow t=4$ or $t=-1/2$$\therefore x=3^4,3^{-1/2}$