Question

Solve the following inequalities.
$lo{g}_{1/3}(x-1)-lo{g}_{1/3}(2x-3)<0.$

Answer 1

${\log }_{1/3}(x-1)-{\log }_{1/3}(2x-3)<0$

$⟹{\log }_{1/3}(x-1)<{\log }_{1/3}(2x-3)$

$⟹x-1>2x-3$

$⟹x<2$

For the $\log$ to be defined-

$x-1>0⟹x>1$

and $2x-3>0⟹x>\frac{3}{2}$

Hence , overall solution is $x\in (\frac{3}{2},2)$