Question

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

Answer 1

Answer: 1

${\log }_4(x-1)={\log }_2(x-3)$
$\Rightarrow \frac{1}{2}{\log }_2(x-1)={\log }_2(x-3)$
$\Rightarrow {\log }_2(x-1{)}^{1/2}={\log }_2(x-3)$
$\Rightarrow (x-1{)}^{1/2}=x-3$
$\Rightarrow x-1=x^2+9-6x$
$\Rightarrow x^2-7x+10=0$
$\Rightarrow (x-2)(x-5)=0$
$\Rightarrow x=2,5$
But $x\ne 2$ because it is not satisfying the domain of given equation i.e ${\log }_2(x-3)\to$ its domain $X>3$
finally $x$ is $5$
$\therefore$ No. of solutions $=1$