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# Tetra-logarithms

The tetra-logarithm (super-logarithm) with base b is the inverse function of the tetra-exponential (super-exponential) with base b.

Just as the two exponentiation functions (powers and exponentials) have two inverse functions (roots and logarithms), the two tetration functions (tetra-powers and tetra-exponentials, also called super-powers and super-exponentials) have two inverse functions (tetra-roots and tetra-logarithms, also called super-roots[1] and super-logarithms[2]).

The base b tetra-logarithm (super-logarithm)[3][4]

$x = b \downarrow\downarrow y = \operatorname{slog}_{b} \, y,\quad b > 0,\, b \ne 1,\, y \in \R^+,$

is the logarithmic-like inverse of the tetra-exponential

$y = b \uparrow\uparrow x = \operatorname{sexp}_{b} \, x,\quad b > 0,\, b \ne 1,\, x \in \R.$

The natural tetra-logarithm (natural super-logarithm)

$x = e \downarrow\downarrow y = \operatorname{slog} \, y,\quad y \in \R^+,$

is the [natural] logarithmic-like inverse of the [natural] tetra-exponential

$y = e \uparrow\uparrow x = \operatorname{sexp} \, x,\quad x \in \R.$

If x is a positive integer, for example, consider the tower of height 4

$2 \uparrow\uparrow 4 = 2^{2^{2^{2}}} = 65536,$

then

$2 \downarrow\downarrow 65536 = \operatorname{slog}_{2} \, 65536 = 4,$

or consider the tower of height 5

$e \uparrow\uparrow 5 = e^{e^{e^{e^{e}}}},$

then

$e \downarrow\downarrow e^{e^{e^{e^{e}}}} = \operatorname{slog} \, e^{e^{e^{e^{e}}}} = 5.$

For a tower of height x = 0, we have

$0 = b \downarrow\downarrow 1 = \operatorname{slog}_{b} \, 1,\quad b > 0,\, b \ne 1,$

and

$1 = b \uparrow\uparrow 0,\quad b > 0,\, b \ne 1,$

since we have the empty product.

## Iterated logarithm

Main article page: 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

$\log^* x := \begin{cases} 0 & \text{if } x \le 1, \\ 1 + \log^*(\log x) & \text{if } x > 1. \end{cases}$

It is thus given by the ceiling of the natural tetra-logarithm (super-logarithm), i.e.

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

The base b iterated logarithm is defined as the number of iterations of the base $b,\, b > 0,\, b \ne 1,$ logarithm before the result is less than or equal to 1, i.e.

$\log^*_b x := \begin{cases} 0 & \text{if } x \le 1, \\ 1 + \log^*_b(\log_b x) & \text{if } x > 1. \end{cases}$

It is thus given by the ceiling of the base b tetra-logarithm, i.e.

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

One might want to find if there is a base β such that the iteration ends exactly at x = 1, by trying tetra-roots of height h, for h = {2, 3, ...}. In that case

$\log^*_{\beta} x = \operatorname{slog}_{\beta} \, x.$

Note that we only have a countable infinity of integer heights h to try... (while there are an uncountable infinity of real numbers greater than 1).

#### Hierarchical list of operations pertaining to numbers [5] [6]

##### 1st iteration
• Addition, S(S(... s times ...(S(n)))), the sum n+s
• Subtraction, P(P(... s times ...(P(n)))), the difference n-s
##### 5th iteration
• Pentation (d as "dimension", b as "base", n as "variable")
• Penta-powers, n^^(n^^(... d times ...(n^^(n^^(n))))), written n^^^d or n↑↑↑d
• Penta-exponentials, b^^(b^^(... n times ...(b^^(b^^(b))))), written b^^^n or b↑↑↑n
• Pentation inverses
##### 6th iteration
• Hexation (d as "dimension", b as "base", n as "variable")
• Hexa-powers, n^^^(n^^^(... d times ...(n^^^(n)))), written n^^^^d or n↑↑↑↑d
• Hexa-exponentials, b^^^(b^^^(... n times ...(b^^^(b)))), written b^^^^n or b↑↑↑↑n
• Hexation inverses
##### 7th iteration
• Heptation (d as "dimension", b as "base", n as "variable")
• Hepta-powers, n^^^^(n^^^^(... d times ...(n^^^^(n)))), written n^^^^^d or n↑↑↑↑↑d
• Hepta-exponentials, b^^^^(b^^^^(... n times ...(b^^^^(b)))), written b^^^^^n or b↑↑↑↑↑n
• Heptation inverses
##### 8th iteration
• Octation (d as "dimension", b as "base", n as "variable")
• Octa-powers, n^^^^^(n^^^^^(... d times ...(n^^^^^(n)))), written n^^^^^^d or n↑↑↑↑↑↑d
• Octa-exponentials, b^^^^^(b^^^^^(... n times ...(b^^^^^(b)))), written b^^^^^^n or b↑↑↑↑↑↑n
• Octation inverses

## Notes

1. Super-rootWikipedia.org.
2. Super-logarithmWikipedia.org.
3. There is no standard or generally accepted notation for the tetra-logarithm yet, although the down-arrow notation (derived from Knuth's up-arrow notation) seems the most intuitive one.
4. Weisstein, Eric W., Down Arrow Notation, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/DownArrowNotation.html]
5. HyperoperationWikipedia.org.
6. Grzegorczyk hierarchyWikipedia.org.