Solving $x$ for $y = x^x$ using a normal scientific calculator (no native Lambert W function)?

Question

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?

$$x=\frac{\ln(y)}{W(\ln y)}= ???$$

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A photo of a primitive calculator caught in the wild:

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

Answer 1

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 $x\approx1.55961047$.