You don’t solve the original equation, which of course cannot be solved algebraically because it is a transcendent equation. You only use the notation of the inverse function f(x) = x e^x, which you call W(x) (the omega function) and which is actually equally unknown, to express the solution. But in this case, what is the difference in directly using the inverse function g(x) of the function f(x) = x^x to say that the solution is x = g(3)? It seems to me that there would be no difference. Regarding the approximate solution, you don’t show any procedure to obtain it. You could have shown, for example, the Newton-Raphson method.

My calculator does not calculate the Lambert-W function! So how do I calculate W(ln(3))?

How do you solve x=1/e^x

Very bad hand writing