TITLE: Solving $x$ for $y = x^x$ using a normal scientific calculator (no native Lambert W function)?
QUESTION [5 upvotes]: Solving $x$ for $y = x^x$ using Lambert W function is clear enough thanks to this handy answer, but as I'm using the solution in a network support document I need it in a form that can be solved on a normal scientific calculator (assume y is a positive integer). People using the document or perhaps embedding the equation in a script might not have real-time access to WolframAlpha.
So my question is: Can $x$ be solved for $y = x^x$ using only the buttons provided on a normal scientific calculator? If so, how?
$$x=\frac{\ln(y)}{W(\ln y)}= ???$$

A photo of a primitive calculator caught in the wild:

(Incidentally, has anyone else ever read "WolframAlpha" and been reminded of "Wolfram & Hart"?)

REPLY [5 votes]: For the sake of my sanity, let's assume $y>1$ to avoid branches and multiple possible answers for $x$.
Applying Newton's method, we have
$$x_{n+1}=x_n+\frac{y(x_n)^{-x_n}-1}{\ln(x_n)+1}$$
For example, take $y=2$, $x_0=1.5$,
$x_1=1.5+\frac{2(1.5)^{-1.5}-1}{\ln(1.5)+1}=1.56308382$
$x_2=x_1+\frac{2(x_1)^{-x_1}-1}{\ln(x_1)+1}=1.559621837$
$x_3=1.55961047$
Thus, the solution to $2=x^x$ is $x\approx1.55961047$.