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Tetra-logarithms

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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.

Contents

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).

See also

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

0th iteration
1st iteration
  • Addition, S(S(... s times ...(S(n)))), the sum n+s
  • Subtraction, P(P(... s times ...(P(n)))), the difference n-s
2nd iteration
3rd iteration
4th iteration
5th iteration
6th iteration
7th iteration
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.

Notes

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