I am looking for a proof, that the equation $$(0.5-t)^{(1+2t)^x}+(0.5+t)^{(1-2t)^x}=1$$ has exactly one solution $x>0$ for every fixed $t\in ]0;0.5[$ .
These solutions $x$ have (seen by numerical tests) a supremum ($t\to 0.5^-$) and an infimum ($t\to 0^+$).
It seems to be, that the infimum is about $\enspace 0.373501\enspace$ and I speculate that the supremum is $1$ .
Has anyone an idea how to express the infimum by a limit value representation, to proof that the supremum must be $1$ or that the transformation $t\to x:=f(t)$, where $f(t)$ is the solution of the equation for every given valid $t$, is bijective.