fun constants 2: r

Let r be the unique real number such that (r-1)^{r} = r. r\approx2.4413754618570030342085445640326490248139472302… or so.
There is an elegant relationship between r and the lambert W function:
observe that if
(r-1)^{r} = r
Then
(r-1)^{r} = (r-1)^{(r-1)^{r}}
Which means that the infinite tetration of (r-1) is r, and
the infinite tetration or power tower is expressible in terms of the Lambert W function.

So we’ve got:
r = - \frac{W(-\log(r-1))}{\log(r-1)}
Now, if we move that r to the other side, we get:
1 = - \frac{W(-\log(r-1))}{r \log(r-1)}
since (r-1)^{r} =r, then latex r\log(r-1} = \log{r}$, so we
can rewrite the bottom like this:

1 = - \frac{W(-\log(r-1))}{\log(r)}
Then:
\log(r) = - W(-\log(r-1))
Then
r = e^{-W(-\log(r-1))}

But since $latex (r-1)^{r} =r, then
r \log(r-1) = \log(r) and finally -\log(r-1) = \log\left(\frac{1}{\sqrt[r]{r}}\right)

So, we’ve got
r = e^{-W(\log\left(\frac{1}{\sqrt[r]{r}}\right))}

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