TITLE: Real values of a function involving the Lambert $W(x)$ function
QUESTION [1 upvotes]: 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.

REPLY [0 votes]: 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)
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