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Tetra-logarithms

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Somnolence that is sloggy[1] slumbering.—Chaucer.
The base
b
tetra-logarithm (super-logarithm) of
x ∈ ℝ+
, notated
slogb x
, is the inverse function of the base
b
tetra-exponential (super-exponential) of
x ∈ ℝ+
, notated
sexpb x
. The natural (base
e
) tetra-logarithm (super-logarithm) of
x ∈ ℝ+
, notated
slog x := sloge x
, is the inverse function of the natural (base
e
) tetra-exponential (super-exponential) of
x ∈ ℝ+
, notated
sexp x := sexpe x
. Indeed, for
x ∈ ℝ+
, the
slog
function is monotonically increasing, but at an astonishingly sloggy pace...

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[2] and super-logarithms[3]).

Contents

The base
b
tetra-logarithm (super-logarithm)[4][5]
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 ≠ 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 [6] [7]

0th iteration
1st iteration
  • Addition:
    S(S( "a times" (S(n))))
    , the sum
    n + a
    , where
    n
    is the augend and
    a
    is the addend. (When addition is commutative both are simply called terms.)
  • Subtraction:
    P(P( "s times" (P(n))))
    , the difference
    ns
    , where
    n
    is the minuend and
    s
    is the subtrahend.
2nd iteration
3rd iteration
4th iteration
5th iteration
6th iteration
7th iteration
8th iteration
  • Octation (
    d
    as "degree",
    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. (obsolete) sluggish
  2. Super-rootWikipedia.org.
  3. Super-logarithmWikipedia.org.
  4. 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.
  5. Weisstein, Eric W., Down Arrow Notation, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/DownArrowNotation.html]
  6. HyperoperationWikipedia.org.
  7. Grzegorczyk hierarchyWikipedia.org.
  8. There is a lack of consensus on which comes first. Having the multiplier come second makes it consistent with the definitions for exponentiation and higher operations. This is also the convention used with transfinite ordinals:
    ω × 2 := ω + ω
    .
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