Explanation

(c)
For the given equation to be meaningful, we must have $x > 0.\ For\ x > 0$, the given equation can be written as
$\frac{3}{4}(\log_{2}{{x)}^{2}\log_{2}{x - \frac{5}{4} = \log_{x}{\sqrt{2} = \frac{1}{2}\log_{x}2}}}$
Put $t = \log_{2}x$ so that $\log_{x}{2 = \frac{1}{t}}$
$\therefore\ \frac{3}{4}t^{2} + t - \frac{5}{4} = \frac{1}{2}\left( \frac{1}{t} \right)$
$\Rightarrow 3t^{3} + 4t^{2} - 5t - 2 = 0$
$\Rightarrow (t - 1)(t + 2)(3t + 1) = 0$
$\Rightarrow \log_{2}{x = t = 1,\ - 2,\ - \frac{1}{3}}$
$\Rightarrow x = 2,\ 2^{- 2},\ 2^{- \frac{1}{3}}$
or $x = 2,\frac{1}{4},\ \frac{1}{2^{1/3}}$
Thus, the given equation has exactly three real solution out of which exactly one is irrational $ie,\frac{1}{2^{1/3}}$.