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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

## Contents

## Iterated natural logarithm and natural tetra-logarithm

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

Thus, if

then

which one might hopefully generalize to (if there could be such a thing as a well-defined real height )

A definition for should at least be strictly increasing, continuous, and have with preferably a continuous first derivative, although it should much preferably be smooth (i.e. all derivatives continuous) since that is the case for whose derivatives are^{(Clarify: Any hint about what should be the derivatives of $\operatorname {slog} \,x$? Maybe that could lead to $\operatorname {slog} \,x$.)} ^{[1]}

One would hope that the smoothness requirement might lead to a unique definition for .

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

lead to a provably unique solution, namely the Gamma function, would then the recursive condition for the tetration

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 **iterated logarithm** is defined as the number of iterations of the base logarithm before the result is less than or equal to 1, i.e.

### Iterated base *b* logarithm and base *b* tetra-logarithm

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

Thus, if

then

which one might hopefully generalize to (if there could be such a thing as a well-defined real height )

## See also

## Notes

- ↑ Needs clarification (Any hint about what should be the derivatives of ? Maybe that could lead to .).— Daniel Forgues 02:20, 27 May 2013 (UTC)