$${\log _4}(x - 1) = {\log _2}(x - 3)$$

$$ \Rightarrow {1 \over 2}{\log _2}(x - 1) = {\log _2}(x - 3)$$

$$ \Rightarrow {\log _2}{(x - 1)^{1/2}} = {\log _2}(x - 3)$$

$$ \Rightarrow {(x - 1)^{1/2}} = {\log _2}(x - 3)$$

$$ \Rightarrow {(x - 1)^{1/2}} = x - 3$$

$$ \Rightarrow x - 1 = {x^2} + 9 - 6x$$

$$ \Rightarrow {x^2} - 7x + 10 = 0$$

$$ \Rightarrow (x - 2)(x - 5) = 0$$

$$ \Rightarrow x = 2,5$$

But x $$ \ne $$ 2 because it is not satisfying the domain of given equation i.e. log₂(x $$-$$ 3) $$ \to $$ its domain x > 3

finally x is 5

$$ \therefore $$ No. of solutions = 1.