Complex Chicanery

I came across an interesting video on YouTube, uploaded today on a channel named Higher Mathematics that posed the following problem:$$\begin{gathered} \text{Solve for }x: \\ {1}^x=5 \end{gathered}$$Clearly the problem has no solution amongst the real numbers but there turns out to be an infinity of solutions once we introduce complex numbers. The solution (with $k=1,2,3\dots$ ) then unfolds:$$\begin{array}{rl}{e}^{i\theta }& =\cos \theta +i\sin \theta \\ e^{i2k\pi }& =\cos 2k\pi +i\sin 2k\pi \\ & =1\\ \therefore (e^{i2k\pi )}{)}^x& =5\\ e^{i2k\pi x}& =5\\ \ln e^{i2k\pi x}& =\ln 5\\ i2k\pi x& =\ln 5\\ x& =\frac{\ln 5}{i2k\pi }\\ & =\frac{-i\ln 5}{2k\pi }\end{array}$$The complex numbers for values of x given by $k=1,2,3$ are shown in Figure 1. They all map to the point $5+0i$.

Figure 1

Thus we have a function $f(x)$ with its domain D being the countably infinite set of points:$$((0,{\displaystyle \frac{-i\ln 5}{2k\pi }})\text{ where k=1, 2, 3, ...})$$ that maps all these points to $(5,0i)$ so that:$$f(x)\to (5,0i)\text{ for all x in D}$$

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Let's look now at a slightly different problem:$$2^x=x$$Right from the start, it can be said once again that there are no real solutions to this equality. This is clear once we graph the line $y=2^x$ and $y=x$ where we see that there are no points of intersection. See Figure 2.

Figure 2

This problem was posed on the same YouTube channel mentioned earlier. Here is a link to the video. So how do we find if any complex numbers satisfy this equality? Firstly, we take the natural logarithms of both sides and proceed from there:$$\begin{array}{rl}{2}^x& =x\\ \ln 2^x& =\ln x\\ x\ln 2& =\ln x\\ \frac{\ln x}{x}& =\ln 2\\ \frac{\ln x}{e^{\ln x}}& =\ln 2\\ \ln x{e}^{-\ln x}& =\ln 2\\ -\ln x{e}^{-\ln x}& =-\ln 2\\ W(-\ln x{e}^{-\ln x})& =W(-\ln 2)\\ -\ln x& =W(-\ln 2)\\ \ln x& =-W(-\ln 2)\\ x& =e^{-W(-\ln 2)}\\ x& =\frac{1}{e^{W(-\ln 2)}}\\ x& \approx -0.37927-0.72087i\end{array}$$where W is the Lambert W function that I've written about in previous posts. See The Omega Constant and the Lambert W Function (June 24th 2020) and More on the Lambert W Function (February 16th 2021). To evaluate $x$ using Wolfram Alpha the command ProductLog[-ln(2)] needs to be used.