I am currently doing things related to the [Akra–Bazzi theorem](http://en.wikipedia.org/wiki/Akra%E2%80%93Bazzi_method). 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.



Transcendence of solutions of $\sum_{i=1}^n a_i b_i^x = 1$
Schanuel's conjecture implies that if $x$ is an algebraic irrational, 
$n^x$ for natural numbers $n$ are linearly independent over the rationals:
see Will Sawin's answer to [this recent question](https://mathoverflow.net/questions/204064/for-what-real-t-is-nt-n-geq-1-linearly-independent-over-mathbbq/204075#204075) for a proof. It's easy to extend this to positive rationals. Thus Schanuel implies that if the $a\_i$ and $b\_i$ are rational, $x$ will either be rational or transcendental. 
So (if you're willing to accept Schanuel) you just have to check for rational solutions, which for something like $(1/3)^x + (3/4)^x = 1$ is not hard.


