How to solve $a^x+b^x=1$ (solve for $x$) I am sorry if this is too easy question for this site, but I really can't find the solution...
$a^x+b^x=1$

Tyma Gaidash asked for context:
I don't have much to add, I was trying to understand how fast population grow assuming that everyone born exactly same amount of time (for example everyone have 2 children 1 born 40 years after the parent born and the other 20 years after the parent born, is it faster than having a twins after 28 years)
If I assume the population formula is $a^x$ and I need to find a
I get that the time it will take to the population multiply itself is the solution to the equation:
$\sum_{i=1}^{n}2^{-t_i/x}=1$ where $t_i$ it after how many year children I born after his parent.
I hoped that there is a way to solve this equation (at least for 2 children ) but I haven't found one.

$$a^x+b^x=1$$
Substitute $x=\frac{\ln(t)}{\ln(a)}$:
$$a^{\frac{\ln(t)}{\ln(a)}}+b^{\frac{\ln(t)}{\ln(a)}}=1$$
$$t+t^{\frac{\ln(b)}{\ln(a)}}=1$$
Substitute $\frac{\ln(b)}{\ln(a)}=\alpha$:
$$t+t^\alpha=1$$
For rational $\alpha$, this equation is related to an algebraic equation and we can use the known solution formulas and methods for algebraic equations. For certain $a,\beta$ and $b=a^\beta$ (WolframAlpha), $\alpha$ is rational.
For rational $\alpha\neq 0,1$, the equation is related to a trinomial equation.
For real or complex $\alpha\neq 0,1$, the equation is in a form similar to a trinomial equation. A closed-form solution can be obtained using confluent Fox-Wright Function $\ _1\Psi_1$ therefore.
see also: How to isolate $x$ in $a^x + b^x = c$? (For use in medical statistics)
Szabó, P. G.: On the roots of the trinomial equation. Centr. Eur. J. Operat. Res. 18 (2010) (1) 97-104
Belkić, D.: All the trinomial roots, their powers and logarithms from the Lambert series, Bell polynomials and Fox–Wright function: illustration for genome multiplicity in survival of irradiated cells. J. Math. Chem. 57 (2019) 59-106
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If $a,b,x\in\mathbb{N}$, your equation is a Diophantine equation.
