Question

The sum of the roots of the equation $3^{{\log }_{\sqrt{2}}x}-12\cdot 3^{{\log }_{2}x}+27=0$ is

• A. $4$
• B. $6$
• C. $8$
• D. $12$

Answer 1

The correct option is B $6$

$3^{{\log }_{\sqrt{2}}x}-12\cdot 3^{{\log }_{2}x}+27=0$
$\Rightarrow 3^{2{\log }_{2}x}-12\cdot 3^{{\log }_{2}x}+27=0$
Put $3^{{\log }_{2}x}=t$
Then, $t^2-12t+27=0$
$\Rightarrow (t-3)(t-9)=0$
$\Rightarrow t=3,9$
$\Rightarrow 3^{{\log }_{2}x}=3$ or $3^{{\log }_{2}x}=9$
$\Rightarrow {\log }_{2}x=1$ or ${\log }_{2}x=2$
$\Rightarrow x=2$ or $x=4$