@rickS, the new equation is $\large\left(e^{ix}\right)^2=\large\left(e^{x} \right)^2+b^2$, solve for $x$.
Now we have a very difficult problem. Whole courses a devoted to the study of theory of equations. And courses on numerical methods.
Here is a sample from the above: $\left(e^{ix}\right)^2=\left(e^{i(2x)}\right)=\cos (2x)+{\bf{i}} \sin(2x)$
I hope you can the difficulty solving for $x$ in: $\cos(2x)+{\bf{i}} \sin(2x)=e^{2x}+b^2~?$