Real values of a function involving the Lambert $W(x)$ function I have the following function:
$$y=-\dfrac{W\left(-\ln(k)\right)}{\ln(k)}$$ 
where $W(x)$ is the Lambert $W$ function defined as the solution of the equation:
$$x=W(x)e^{W(x)}$$
If $k\in\mathbb{R}$ and $k\gt 0$, for what values of $k$, $y$ is real?
I found heuristically that the value of $k$ should be in the range $1.44\lt k\lt 1.45$. How can I found it analitically, or how can I get the best approximation for the maximum value of $k$ for which the solution is real? Thanks.
 A: The inverse function of $y$ is given by $\left(\frac{1}{y}\right)^{-\frac{1}{y}}$, hence it is enough to locate the maximum of $g(x)=x^{-x}$ over $\mathbb{R}^+$ to get that the domain of $y$ is given by 
$$ \left(0,\left(\frac{1}{e}\right)^{-\frac{1}{e}}\right).$$
A: 
Notice, for the 'Product Log Function' (or the Lambert $\text{W}$-function) is defined as follows:
$$f(z)=ze^z\to z=f^{-1}(ze^z)=\text{W}(ze^z)$$


Now, we know that for $k\in\mathbb{R}^+$ (real, and bigger than zero) $\ln(k)$ is well defined.
Now, your question is about:
$$y(k)=k^{k^{k^{k^{\dots}}}}=-\frac{\text{W}\left(-\ln(k)\right)}{\ln(k)}$$
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).
So, the domain of $y(k)$:
$$\left[0,e^{\frac{1}{e}}\right]$$
A: The Lambert W function has two real branches.  For $W_0$, your domain is $(0,\exp(1/e))$ as noted by Jack.  For $W_{-1}$, your domain is $(1,\exp(1/e))$.

$y=-\dfrac{W_0\left(-\ln(k)\right)}{\ln(k)}$ (red) and $y=-\dfrac{W_{-1}\left(-\ln(k)\right)}{\ln(k)}$ (blue)
