The number of solutions of log_4 (x-1)=log_2 (x-3), is _____.

For the given equation to be valid, we must have 

$x-1 > 0$ and $x-3>0⇒x> 3$

Now,

$\log_4 (x-1)=\log_2 (x-3)$

$⇒\frac{1}{2}\log_2 (x-1)=\log_2 (x-3)$

$⇒\log_2 (x-1)=\log_2 (x-3)^2$

$⇒x-1=(x-3)^2$

$⇒x^2-7x+10=0⇒x=2,5⇒ x=5$   $[∵ x>3]$