Iterated logarithm

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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:

• A (probably wrong!) definition for a real height η, with 0 ≤ η < 1;
• A (better?) definition for a real height w, with 0 ≤ w < 1.

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 $\eta ,\,0\leq \eta <1$)

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

See also

• {{log-star}} (mathematical formatting template)
• {{log^*}} (mathematical function template)