Question

Find the values of x which satisfying the equation $(lo{g}_{\sqrt{3}}x-2)-(lo{g}_{3}x-2)=2$.

Answer 1

${\log }_{\sqrt{3}}x-2-{\log }_{3}x+2=2$

$\Rightarrow {\log }_{3^{\frac{1}{2}}}x-{\log }_{3}x=2$

$\Rightarrow 2{\log }_{3}x-{\log }_{3}x=2\dots ({\log }_{b^x}a=\frac{1}{x}{\log }_{b}a)$

$\Rightarrow {\log }_{3}x=2$

We know that, ${\log }_{a}b=c⟹b=a^c$

$\Rightarrow x=3^2=9$