Question

The number of solutions of the equation ${\log }_4(x-1)={\log }_2(x-3)$ is

Answer 1

${\log }_4(x-1)={\log }_2(x-3)$
$\Rightarrow \frac{1}{2}{\log }_2(x-1)={\log }_2(x-3)$
$\Rightarrow x-1=(x-3{)}^2$
$\Rightarrow x^2-6x+9=x-1$
$\Rightarrow x^2-7x+10=0$
$\Rightarrow x=2,5$
$x=2$ is not possible as ${\log }_2(x-3)$ is not defined at $x=2.$
$\therefore$ Number of solution $=1$