Iterating $\log(x\log(x\log(...)))$

Question

For a real positive $x$, let $F(x)$ denote the sequence $$ \left(\log x,\log(x\log(x)),\log(x\log(x\log(x))),\log(x\log(x\log(x\log(x)))),...\right), $$ stopping at the first nonpositive value or infinite if it keeps staying positive.

Trivially $F(x)=(\log x)$ for $x\leqslant1$, and $F(e)=(1,1,1,1,...)$.

What else can be said? Is $F(x)$ finite for any $x<e$? Is it infinite for any $x\geqslant e$? Can you compute the limiting value for any infinite $F(x)$ except $F(e)$?

To illustrate the answer and comments below it, here is a plot:

The Mathematica code used to produce that plot:

Plot[{Labeled[-LambertW[-1/x], -W[-1/"x"]],
      Labeled[Log[x], Log["x"]],
      Labeled[Log[x Log[x]], Log["x" Log["x"]]],
      Labeled[Log[x Log[x Log[x]]], Log["x" Log["x" Log["x"]]]]
     }, {x, E, 10}]

Answer 1

For $x>e,$ $\ln(x)>1$ then let's define the sequence $f_0(x)=x,$ $f_{n+1}(x)=\ln(xf_{n}(x)) $

to prove the limit $\displaystyle \lim_{n\to\infty} f_n(x)$ always exist for any $x>e$ I will use monotone convergence theorem

$\boxed{\textbf{Monotone convergence theorem}:-\text{A monotone sequence of Real numbers converge if it bounded}}$

if you want to see the proof of this theorem see this

First, I will prove that the sequence is monotone: since $\ln(x\ln(x))> \ln(x)$ or in other words $f_2(x)> f_1(x)$ assume that $f_k(x)> f_{k+1}(x)$ then $f_{k}(x) =\ln(x f_{k-1}(x))$ and $f_{k+1}(x) =\ln(x f_{k}(x))$ and since $\ln(a)> \ln(b)$ if $a>b$ we conclude by induction that the sequence is monotone increasing i.e $f_{n+1}(x)> f_{n}(x)\ \forall n \in \mathbb{N}$.

Secondly, I will prove the sequence is bounded for any $x$: since $\sqrt x > \ln(x)$ and $\sqrt a > \sqrt b $ if $a>b $, then we define the sequence $g_{0}(x) =x ,g_{n+1}(x)=\sqrt{xg_n(x)}$ or in other form $g_n(x)= \displaystyle x^{\sum_{k=1}^n \frac{1}{2^k}}$ and it is obvious that $g_n(x)> f_n(x) \forall n \in \mathbb{N}$ and that $\displaystyle \lim_{n \to \infty} g_n(x)=x $ so the sequence is bounded above by $x$.

By Monotone convergence theorem, the limit $\displaystyle \lim_{n\to\infty} f_n(x)$ exist and lets define $\displaystyle F(x):=\lim_{n\to\infty} f_n(x)$which mean that we can write

$$F(x)= \ln(x F(x))$$ $$e^{F(x)}= x F(x)$$ $$\frac{-1}{x}= -F(x)e^{-F(x)}$$ $$F(x)= -\operatorname{W}_{-1}\left( \frac{-1}{x}\right)$$

where $\operatorname{W}$ denotes Lambert W function

as @Gary pointed out in the comments referring to 4.13.11 , in NIST Digital Library of Mathematical Functions $W_{−1}$ is the unique branch of $W$ that stays real between $−\frac{1}{e}$ and $0$ and monotonely decreases ز