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 Somnolence that is sloggy^{[1]} slumbering.—Chaucer.
The base
tetralogarithm (
superlogarithm) of
, notated
, is the inverse function of the base
tetraexponential (
superexponential) of
, notated
. The natural (base
)
tetralogarithm (
superlogarithm) of
, notated
, is the inverse function of the natural (base
) tetraexponential (superexponential) 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 (tetrapowers and tetraexponentials, also called superpowers and superexponentials) have two inverse functions (tetraroots and tetralogarithms, also called superroots^{[2]} and superlogarithms^{[3]}).
The base
tetralogarithm (superlogarithm)
^{[4]}^{[5]}

is the logarithmiclike inverse of the tetraexponential

The natural tetralogarithm (natural superlogarithm)

is the [natural] logarithmiclike inverse of the [natural] tetraexponential

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

then

or consider the tower of height

then

For a tower of height
, we have

and

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

It is thus given by the ceiling of the natural tetralogarithm (superlogarithm), i.e.

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.

It is thus given by the
ceiling of the base
tetralogarithm, i.e.

One might want to find if there is a base
such that the iteration ends exactly at
, by trying
tetraroots of height
, for
. In that case

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").
 Octapowers:
n ^^^^^ (n ^^^^^ (… "d times" … (n ^^^^^ (n)))) 
, written .
 Octaexponentials:
b ^^^^^ (b ^^^^^ (… "n times" … (b ^^^^^ (b)))) 
, written .
 Octation inverses
Notes
 ↑ (obsolete) sluggish
 ↑ Superroot—Wikipedia.org.
 ↑ Superlogarithm—Wikipedia.org.
 ↑ There is no standard or generally accepted notation for the tetralogarithm yet, although the downarrow notation (derived from Knuth's uparrow notation) seems the most intuitive one.
 ↑ Weisstein, Eric W., Down Arrow Notation, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/DownArrowNotation.html]
 ↑ Hyperoperation—Wikipedia.org.
 ↑ Grzegorczyk hierarchy—Wikipedia.org.
 ↑ 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: .