$$\frac{3 (1 - 2x)}{{\sqrt[4]{(1 - x)x}}} = \log_2\left(\frac{1 - x}{x}\right) \sqrt {2} $$ It is clear the $\frac{1}{2}$ is a solution, but there are others solutions. If we subtract this two side of equation and plot in a visualization software, we can see that are others roots. I like to find the solution that are close to $x = 0.2$.
Addendum: I need a exact solution.
As you did, consider that you look for the zero's of function $$f(x)=\frac{3 (1 - 2x)}{{\sqrt {2}} {\sqrt[4]{(1 - x)x}}} - \log_2\left(\frac{1 - x}{x}\right)$$ Beside the trivial $x=0.5$, graphing, you noticed three other roots close to $x=0.2$, $x=0.8$ and another close to $x=1$; if you plot again the function between $0.95$ and $1.0$, the largest root is close to $x=0.99$.
Now, using these values a starting points, use Newton method and get the following iterates $$\left( \begin{array}{cc} n & x_n \\ 0 & 0.2000000000 \\ 1 & 0.1704681790 \\ 2 & 0.1750941042 \\ 3 & 0.1752348886 \\ 4 & 0.1752350156 \end{array} \right)$$
$$\left( \begin{array}{cc} n & x_n \\ 0 & 0.8000000000 \\ 1 & 0.8295318210 \\ 2 & 0.8249058958 \\ 3 & 0.8247651114 \\ 4 & 0.8247649844 \end{array} \right)$$
$$\left( \begin{array}{cc} n & x_n \\ 0 & 0.9900000000 \\ 1 & 0.9912580691 \\ 2 & 0.9911109005 \\ 3 & 0.9911082777 \\ 4 & 0.9911082769 \end{array} \right)$$