If $x$ is rational, can $\log(1-x)/\log x$ be algebraic?

Question

If $x$ is positive rational number less than $\frac{1}{2}$, can the following logarithmic expression be equivalent to real algebraic number, say $g$?

$$\frac{\log(1-x)}{\log x} = g$$

Answer 1

Your identity gives: $$ 1-x = x^g \tag{1}$$ where $x\in\mathbb{Q}$ trivially gives that the LHS is a rational number. If $g$ is not a rational number, the Gelfond-Schneider theorem gives that the RHS is a trascendental number, contradiction.

So $g$ has to be a rational number. But in order that $1-x$ and $x^g$ are rational numbers with the same denominator, $g$ has to be one. So $x=\frac{1}{2}$ and $g=1$ is the only solution.