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][1] 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.


  [1]: http://mathoverflow.net/questions/204064/for-what-real-t-is-nt-n-geq-1-linearly-independent-over-mathbbq/204075#204075