Constant Exponent

Typically equations involving both $x$ and $e^x$ don't have analytical solutions (ie solutions in terms of elementary functions). Wolfram|Alpha gives the answer in terms of the Lambert W function, although this is more or less the definition of that function, so it's a bit of a tautology.

The Newton-Raphson approach used in the discussion clearly works well - I wonder if it leads to any interesting fractal structure?

An alternative computational method is to simply iterate the function. Doing this in the original form blows up rapidly - but note that we also have $z=\log z$ . We're taking the principal value of the logarithm in order to find a solution in the specified part of the complex plane.

This is most easily iterated by writing $z=Re^{i \theta}$ . Then $\log z=\log R + i \theta$ (this is a naive approach to the complex logarithm but is good enough here). This gives as a map

$R \rightarrow \sqrt{(\log R)^2+\theta^2}$

and

$\theta \rightarrow \arctan {\frac{\theta}{\log R}}$

Again, a bit of care needs to be taken with the $\arctan$ function when implementing this, but with a reasonable initial estimate it converges quickly.

Incidentally, if you replace the above with $\theta \rightarrow \arctan {\frac{\theta}{\log R}}+2k\pi$ for integer $k$ , you can converge to different roots outside the range you specified (one example is at around $z=2.06227773+7.588631178i$ ).