I am currently doing things related to the Akra–Bazzi theorem. One element in that theorem is the following:

For $n>0$ and sequences of real numbers $a_i, b_i$ of length $n$, where all $a_i>0$ and all $b_i\in (0;1)$, we consider the unique real number $x$ such that $$\sum_{i=1}^n a_i b_i^x = 1$$

My question is: when is $x$ rational/algebraic/transcendental?

For example, for the equation
$$\left(\frac{1}{3}\right)^x + \left(\frac{3}{4}\right)^x = 1$$
$x$ is in $(1.1519623;1.1519624)$. It *looks* like a transcendental number to me, but I have no idea how to show that it is or is not.