The number of solutions of the equation _ 4 (x - 1) = _ 2 (x - 3)...

$⇒ \dfrac{1}{2}\log_{2}\left(x - 1\right) = \log_{2}\left(x - 3\right)$

$⇒ \log_{2}{\left(x - 1\right)}^{1 / 2} = \log_{2}\left(x - 3\right)$

$⇒ {\left(x - 1\right)}^{1 / 2} = x - 3$

$⇒ x - 1 = {x}^{2} + 9 - 6x$

$⇒ {x}^{2} - 7x + 10 = 0$

$⇒ \left(x - 2\right)\left(x - 5\right) = 0$

$⇒ x = 2, 5$

But $x \ne 2$ because it is not satisfying the domain of given equation i.e $\log_{2}\left(x - 3\right) \to$ its domain $x > 3$ finally $x$ is $5$

$∴$ No. of solutions $= 1$.