Solving tetration equation with Lambert W Function

Question

Does anyone know how to use the Lambert W Function to solve the equation $$x^{x^x}=2$$ for $x$?

I've figured out how to solve these two similar equations: $$x^x=2$$ $$x^{x^{x+1}}=2$$

Answer 1

We can use Newton's method to approximate the answer

$x^{x^x}=2 \implies x^{x^x}-2=0 $

Let $f(x)=x^{x^x}$ after a lot of work we get that $f'(x)=x^x\times x^{x^x} \times ({1\over x} + \log x +\log^2x)$

Now let's pick a starting value

$x_1=1$ Then every next value would be of the form $x_{n+1 }=x_n - \frac{f(x_n)}{f'(x_n)}$

then $x_2=2$

$x_3 \approx 1.72256 $

$x_4 \approx 1.58439 $

$x_5 \approx 1.72256 $

...

at around $x_{10} $ it converges to $1.4766843374$

Turns out it is a really good approximation too! As $1.4766843374^{1.4766843374^{1.4766843374}}=2.0000000001826366$

Not really what you probably expected , I am still not sure if this is solvable using the Lambert W function but I am pretty sure it is not .