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Tetra-logarithms

Somnolence that is sloggy 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 and super-logarithms).

The base
 b
tetra-logarithm (super-logarithm)
$x=b\downarrow \downarrow y=\operatorname {slog} _{b}\,y,\quad b>0,\,b\neq 1,\,y\in \mathbb {R} ^{+},$ is the logarithmic-like inverse of the tetra-exponential

$y=b\uparrow \uparrow x=\operatorname {sexp} _{b}\,x,\quad b>0,\,b\neq 1,\,x\in \mathbb {R} .$ The natural tetra-logarithm (natural super-logarithm)

$x=e\downarrow \downarrow y=\operatorname {slog} \,y,\quad y\in \mathbb {R} ^{+},$ is the [natural] logarithmic-like inverse of the [natural] tetra-exponential

$y=e\uparrow \uparrow x=\operatorname {sexp} \,x,\quad x\in \mathbb {R} .$ 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
$0=b\downarrow \downarrow 1=\operatorname {slog} _{b}\,1,\quad b>0,\,b\neq 1,$ and

$1=b\uparrow \uparrow 0,\quad b>0,\,b\neq 1,$ 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
$\log ^{*}x:={\begin{cases}0&{\text{if }}x\leq 1,\\1+\log ^{*}(\log x)&{\text{if }}x>1.\end{cases}}$ 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.
$\log _{b}^{*}x:={\begin{cases}0&{\text{if }}x\leq 1,\\1+\log _{b}^{*}(\log _{b}x)&{\text{if }}x>1.\end{cases}}$ 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  

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  n  −  s
, where  n
is the minuend and  s
is the subtrahend.
2nd iteration
• Multiplication:  n + (n + (⋯ "k times" ⋯ (n + (n))))
, the product  m  ⋅   k
, where  m
is the multiplicand and  k
is the multiplier. (When multiplication is commutative both are simply called factors.)
• Division: the ratio  n  /  d
, where  n
is the dividend and  d
is the divisor.
3rd iteration
• Exponentiation ( d
as "degree",  b
as "base",  n
as "variable").
• Powers:  n  ⋅   (n  ⋅   (⋯ "d times" ⋯ (n  ⋅   (n))))
, written  n d
.
• Exponentials:  b  ⋅   (b  ⋅   (⋯ "n times" ⋯ (b  ⋅   (b))))
, written  b n
.
• Exponentiation inverses ( d
as "degree",  b
as "base",  n
as "variable").
5th iteration
• Pentation ( d
as "degree",  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 "degree",  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 "degree",  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 "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