Power tower convergence outside of convergence bounds, where did I go wrong?

Question

It is said a power tower

$$x^{x^{x^{x^{x^ \cdots}}}}$$

converges iff $ e^{-e} \lt x \lt e^{\frac{1}{e}}$

Is the following incorrect?

Assume $$x^{x^{x^{x^{x^ \cdots}}}} = \frac{1}{1000000}$$ Then we can say $$x^{\frac{1}{1000000}} = \frac{1}{1000000}$$ $$\ln{ x^{\frac{1}{1000000}}} = \ln{ \frac{1}{1000000}}$$ $$\frac{1}{1000000} \ln{x} = \ln{ \frac{1}{1000000}}$$ $$\ln{x} = 1000000 \ln{ \frac{1}{1000000}}$$ $$ x = e^{1000000 \ln{ \frac{1}{1000000}}}$$ Here, $x$ is less than our bound of $ e^{-e}$

Answer 1

Notice, for the 'Product Log Function' (or the Lambert $\text{W}$-function) is defined as follows:

$$f(z)=ze^z\to z=f^{-1}(ze^z)=\text{W}(ze^z)\tag1$$

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Now, we know that for $k\in\mathbb{R}^+$ (real, and bigger than zero) $\ln(k)$ is well defined.

Now, your question is about:

$$y(k)=k^{k^{k^{k^{\dots}}}}=-\frac{\text{W}\left(-\ln(k)\right)}{\ln(k)}\tag2$$

Eisenstein's (1844) considered this series of the infinite power tower. $y(k)$ converges iff $e^{-e}\le k\le e^{\frac{1}{e}}$; OEIS A073230 and A073229), as shown by Euler (1783) and Eisenstein (1844) (Le Lionnais 1983; Wells 1986, p. 35).