I have the following equation:
$$2^{-\frac{1}{x}}+1=-\log_2\left(\frac{1}{x}-1\right)$$
I can't find a way to solve it.
I got it by the function $f(x)=\frac{1}{1+2^{-\frac{1}{x}}}$ when I was finding the point where $x=y$. For this I inversed the function, resulting $f^{-1}(x) = -\frac{1}{\log_2\left(\frac{1}{x}-1\right)}$. Equated and simplified results in the firts equation. I know is simpler to equalize $f(x)$ and $x$ and solve it with the Lambert's W function, but I was wondering about the first one, considering Wolfram Alpha can't solve it step by step (does it solve by testing?) and both equations returns same value for $x$, being something different.
Plot:
As you can see, there is one single point where $x=y$.