Question

The set of all $x$ satisfying the equation $x^{{\log }_3{x}^2+({\log }_{3}x{)}^2-10}=\frac{1}{x^2}$ is

• A. $\{1,9\}$
• B. $\{1,9,\frac{1}{81}\}$
• C. $\{1,4,\frac{1}{81}\}$
• D. $\{9,\frac{1}{81}\}$

Answer 1

The correct option is B $\{1,9,\frac{1}{81}\}$

$x^{{\log }_3{x}^2+({\log }_{3}x{)}^2-10}=\frac{1}{x^2}$
where, $x>0$
$\Rightarrow [{\log }_3{x}^2+({\log }_{3}x{)}^2-10]\log x=-2\log x$

$[{\log }_3{x}^2+({\log }_{3}x{)}^2-10]\log x+2\log x=0$

Taking logx common,

$logx([{\log }_3{x}^2+({\log }_{3}x{)}^2-10]+2)=0$
if $\Rightarrow \log x=0$
$\Rightarrow x=1$ ....(1)
or ${\log }_3{x}^2+({\log }_{3}x{)}^2-10=-2$
$\Rightarrow ({\log }_{3}x{)}^2+2{\log }_{3}x-8=0$
$\Rightarrow {\log }_{3}x=-4,2$
$\Rightarrow x=\frac{1}{81},9$ ...(2)
From (1) & (2)
$x\in \{1,9,\frac{1}{81}\}$
Ans: B