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- Somnolence that is sloggy^{[1]} slumbering.—Chaucer.
The base
tetra-logarithm (
super-logarithm) of
, notated
, is the inverse function of the base
tetra-exponential (
super-exponential) of
, notated
. The natural (base
)
tetra-logarithm (
super-logarithm) of
, notated
, is the inverse function of the natural (base
) tetra-exponential (super-exponential) of
, notated
. Indeed, for
, the
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]}).
The base
tetra-logarithm (super-logarithm)
^{[4]}^{[5]}
- $x=b\downarrow \downarrow y=\operatorname {slog} _{b}\,y,\quad b>0,\,b\neq 1,\,y\in \mathbb {R} ^{+},$
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
is a positive integer, for example, consider the tower of height
- $2\uparrow \uparrow 4=2^{2^{2^{2}}}=65536,$
then
- $2\downarrow \downarrow 65536=\operatorname {slog} _{2}\,65536=4,$
or consider the tower of height
- $e\uparrow \uparrow 5=e^{e^{e^{e^{e}}}},$
then
- $e\downarrow \downarrow e^{e^{e^{e^{e}}}}=\operatorname {slog} \,e^{e^{e^{e^{e}}}}=5.$
For a tower of height
, 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
. 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
iterated logarithm is defined as the number of iterations of the base
logarithm before the result is less than or equal to
, 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
tetra-logarithm, i.e.
- $\log _{b}^{*}x:=\lceil \operatorname {slog} _{b}\,x\rceil .$
One might want to find if there is a base
such that the iteration ends exactly at
, by trying
tetra-roots of height
, for
. In that case
- $\log _{\beta }^{*}x=\operatorname {slog} _{\beta }\,x.$
Note that we only have a countable infinity of integer heights
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]}
0^{th} iteration
1^{st} iteration
- Addition:
S(S(⋯ "a times" ⋯ (S(n)))) |
, the sum , where is the augend and is the addend. (When addition is commutative both are simply called terms.)
- Subtraction:
P(P(⋯ "s times" ⋯ (P(n)))) |
, the difference , where is the minuend and is the subtrahend.
2^{nd} iteration
- Multiplication:
n + (n + (⋯ "k times" ⋯ (n + (n)))) |
, the product , where is the multiplicand and is the multiplier.^{[8]} (When multiplication is commutative both are simply called factors.)
- Division: the ratio , where is the dividend and is the divisor.
3^{rd} iteration
- Exponentiation ( as "degree", as "base", as "variable").
- Powers:
n ⋅ (n ⋅ (⋯ "d times" ⋯ (n ⋅ (n)))) |
, written .
- Exponentials:
b ⋅ (b ⋅ (⋯ "n times" ⋯ (b ⋅ (b)))) |
, written .
- Exponentiation inverses ( as "degree", as "base", as "variable").
4^{th} iteration
- Tetration ( as "degree", as "base", as "variable").
- Tetration inverses ( as "degree", as "base", as "variable").
5^{th} iteration
- Pentation ( as "degree", as "base", as "variable").
- Pentation inverses
6^{th} iteration
- Hexation ( as "degree", as "base", as "variable").
- Hexation inverses
7^{th} iteration
- Heptation ( as "degree", as "base", as "variable").
- Heptation inverses
8^{th} iteration
- Octation ( as "degree", as "base", as "variable").
- Octa-powers:
n ^^^^^ (n ^^^^^ (⋯ "d times" ⋯ (n ^^^^^ (n)))) |
, written .
- Octa-exponentials:
b ^^^^^ (b ^^^^^ (⋯ "n times" ⋯ (b ^^^^^ (b)))) |
, written .
- Octation inverses
Notes