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Iterated logarithm

The iterated natural logarithm, denoted log  (usually read "log star"), is defined as the number of iterations of the natural logarithm before the result is less than or equal to 1. It is defined recursively as

$\log ^{*}x:={\begin{cases}0&{\text{if }}x\leq 1,\\1+\log ^{*}(\log x)&{\text{if }}x>1.\end{cases}}$ Iterated natural logarithm and natural tetra-logarithm

The iterated natural logarithm corresponds to the ceiling of the natural tetra-logarithm (super-logarithm), i.e.

$\log ^{*}x=\lceil \operatorname {slog} \,x\rceil .$ Thus, if

$x=e\uparrow \uparrow \log ^{*}x=\underbrace {{e}^{{.\,}^{{.\,}^{{.\,}^{{e}^{{e}^{{e}^{e}}}}}}}} _{\log ^{*}x},$ then

$\operatorname {slog} \,x=\log ^{*}x,$ which one might hopefully generalize to (if there could be such a thing as a well-defined real height $\eta ,\,0\leq \eta <1$ )

$\operatorname {slog} \,x\,{\overset {?}{=}}\,-\eta +\log ^{*}x,\quad 0\leq \eta <1,\quad x\in \mathbb {R} +,\,x\geq 1.$ A definition for $\operatorname {slog} \,x$ should at least be strictly increasing, continuous, and have $\operatorname {slog} \,1=0,\,\operatorname {slog} \,e=1,$ with preferably a continuous first derivative, although it should much preferably be smooth (i.e. all derivatives continuous) since that is the case for $\log x,\,x\in \mathbb {R} ^{+},$ whose derivatives are (Clarify: Any hint about what should be the derivatives of $\operatorname {slog} \,x$ ? Maybe that could lead to $\operatorname {slog} \,x$ .)

${\frac {{\rm {d}}^{k}\log x}{{\rm {d}}x^{k}}}={\frac {(-1)^{k-1}\,(k-1)!}{x^{k}}},\quad x\in \mathbb {R} ^{+},\,k\geq 1.$ One would hope that the smoothness requirement might lead to a unique definition for $\operatorname {slog} \,x$ .

Also, considering that for the generalization of the factorial, the recursive condition

$f(x)=x\,f(x-1)$ lead to a provably unique solution, namely the Gamma function, would then the recursive condition for the tetration

$e\uparrow \uparrow x=e^{e\uparrow \uparrow (x-1)}$ lead to a provably unique solution for the generalization of tetration?

Tentative definitions for a real height

See on the talk page:

Base b iterated logarithm

The base $b$ iterated logarithm is defined as the number of iterations of the base $b,\,b>1,$ logarithm before the result is less than or equal to 1, i.e.

$\log _{b}^{*}x:={\begin{cases}0&{\text{if }}x\leq 1,\\1+\log _{b}^{*}(\log _{b}x)&{\text{if }}x>1.\end{cases}}$ Iterated base b logarithm and base b tetra-logarithm

The base $b$ iterated logarithm corresponds to the ceiling of the base $b$ tetra-logarithm (super-logarithm), i.e.

$\log _{b}^{*}x=\lceil \operatorname {slog} _{b}\,x\rceil .$ Thus, if

$x=b\uparrow \uparrow \log _{b}^{*}x=\underbrace {{b}^{{.\,}^{{.\,}^{{.\,}^{{b}^{{b}^{{b}^{b}}}}}}}} _{\log _{b}^{*}x},$ then

$\operatorname {slog} _{b}\,x=\log _{b}^{*}x,$ which one might hopefully generalize to (if there could be such a thing as a well-defined real height $\eta ,\,0\leq \eta <1$ )

$\operatorname {slog} _{b}\,x\,{\overset {?}{=}}\,-\eta +\log _{b}^{*}x,\quad 0\leq \eta <1.$ 