Non-existence of closed-form solutions An equation like
$$a^x+b^x=1$$ 
can be turned to the form 
$$t^\alpha+t=1$$ by a suitable change of variable.
When $\alpha$ is a rational we can put that in a polynomial form
$$u^p+u^q=1$$ and use Galois theory to refute the existence of a solution expressible by radicals.
But what about $\alpha$ being irrational, and what about the (non-)existence of closed-form formulas ?

Update:
From the comment by @mercio, we know that for every $\alpha$ of the form $\log_c(1-c)$ where $c$ has a closed form expression, we have the closed-form solution $t=c$. This settles the first part of the question.
So is there a way to characterize the $\alpha$'s for which a closed-form solution exists ?
 A: My answer is for solutions in terms of elementary functions.
Let $\alpha,a,b,t,x\in\mathbb{C}$.
$$t^\alpha+t=1\tag{1}$$
You already treated the case of rational $\alpha$.
$\ $
1.) Solutions in the algebraic numbers
Let $\alpha$ be an irrational algebraic number and $t$ algebraic.
a) $0$ and $1$ aren't solutions of equation 1. Therefore $t\notin\{0,1\}$.
b) Assume that $t\notin\{0,1\}$. Then Gelfond–Schneider theorem implies that $t^\alpha$ is transcendental. Because equation 1 is an algebraic equation of $t^\alpha$ and $t^\alpha$ is transcendental, equation 1 is a contradiction. $t\notin\{0,1\}$ cannot be algebraic therefore.
Because of a) and b), $t$ cannot be algebraic.
2.) Solutions by elementary partial inverse functions
The elementary functions are generated from their complex function variable by applying finite numbers of $\exp$, $\ln$ and/or unary or multiary algebraic functions.
Solving equation 1 by rearranging it by operations we can read from the equation means applying appropriate partial inverse functions of the elementary function $f\colon t\mapsto t^\alpha+t$.
The incomprehensibly unfortunately hardly noticed theorem in [Ritt 1925], that is proved also in [Risch 1979], implies what kinds of elementary functions have elementary partial inverse functions: The elementary functions having elementary partial inverse functions are generated from their complex function variable by applying finite numbers of $\exp$, $\ln$ and/or unary algebraic functions.
We have $f(t)=A(t^\alpha,t)$, wherein $A$ is an algebraic function. For irrational $\alpha$, $\text{trdeg}_{\overline{\mathbb{Q}}}(t^\alpha,t)>0$, and with help of Ritt's theorem we conclude that $f$ doesn't have elementary partial inverse functions. Therefore equation 1 is not solvable by rearranging it by operations we can read from the equation.
[Ritt 1925] Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90
[Risch 1979] Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J. Math. 101 (1979) (4) 743-759
3.) Solutions in the elementary numbers
The elementary numbers are the numbers that are generated from the rational numbers by applying only elementary functions.
For irrational $\alpha$, the problem of solving equation 1 by elementary numbers is unsolved because the only general statement we have for this kinds of problems is Lin's theorem in [Lin 1983]. See e.g. Example equation which does not have a closed-form solution. But Lin's theorem is applicable only to irreducible algebraic equations of simultaneously $z\in\mathbb{C}$ and $e^z$, and equation 1 possibly cannot be transformed into this form.
[Lin 1983] Ferng-Ching Lin: Schanuel's Conjecture Implies Ritt's Conjectures. Chin. J. Math. 11 (1983) (1) 41-50
[Chow 1999] Chow, T.: What is a closed-form number. Am. Math. Monthly 106 (1999) (5) 440-448
