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Penta-logarithms

The "penta-logarithm"

$b\downarrow \downarrow \downarrow n\equiv \log _{b}^{**}(n)=d,\,$ is the logarithmic-like inverse of the "penta-exponential"

$b\uparrow \uparrow \uparrow d\equiv \exp _{b}^{**}(d)=n.\,$ The ceiling of the "penta-logarithm" of a positive integer $n\,$ denotes the number of iterations of the tetra-logarithm (base $b\,$ ) that is required such that

$\underbrace {\log _{b}^{*}\ldots \log _{b}^{*}} _{\lceil d\rceil }n\leq b,\,$ where $d\,=\,\lceil d\rceil \,$ is a nonnegative integer when $n\,$ is of the form

$n=\underbrace {\exp _{b}^{*}\ldots \exp _{b}^{*}} _{d}1,\quad d\geq 0.\,$ It should be possible to generalize $d\,$ to integers, rational numbers, real numbers and complex numbers, as has been done for exponentials and logarithms.

Note: there is no normed or generally accepted notation for the "penta-logarithm" yet, although the down-arrow notation (derived from Knuth's up-arrow notation) seems the most intuitive one.

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