I need to solve an equation of the form $$a^x+b^x=c$$ with $a,b\in (0,1)$ and $c\in(0,2)$ (and I'm solving for $x\in\mathbb{R}_{>0}$).
I know this admits a solution (details below), but it's such a simple equation that I'm sure its properties have been studied before and I need to find that literature. And if this admits a closed form solution, then my summer would be saved! :)
Existence: Since $a,b\in (0,1)$, the left side is continuous strictly decreasing in $x$ for all positive values. Moreover, $a^0+b^0\equiv 2>c$ and $\lim_{x\to\infty} a^x+b^x \equiv 0<c$ for all admissible parameter values. By the intermediate value theorem, $x>0$ must exist. But what is it???
For the most general case, the root of equation $$f(x)=a^x+b^x-c$$ does not show any closed form and numerical methods should be used.
The root is bounded by the roots of $2a^x=c$ and $2b^x=c$ which are simple and then we have a range which contains the solution we look for.
So, a root finder method, such as Newton method could start,a the mid point, using $$x_0=\frac{ \log (\sqrt{ab})}{ \log (a) \log (b)}\log \left(\frac{c}{2}\right)$$ and will converge quite fast taking into account the acceptable ranges $a,b\in (0,1)$, $c\in(0,2)$.
For illustration purposes, consider $a=\frac{1}{3}$, $b=\frac{1}{2}$, $c=1$. So, the successive iterates will be $$x_0=1.15391338000$$ $$x_1=1.12311432825$$ $$x_2=1.12353135181$$ $$x_3=1.12353142978$$ which is the soution for twelve significant figures.