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# Hepta-logarithms

From OeisWiki

The "hepta-logarithm"

is the logarithmic-like inverse of the "hepta-exponential"

The ceiling of the **"hepta-logarithm"** of a positive integer denotes the number of iterations of the hexa-logarithm (base ) that is required such that

where is a nonnegative integer when is of the form

It should be possible to generalize 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 "hepta-logarithm" yet, although the down-arrow notation (derived from Knuth's up-arrow notation) seems the most intuitive one.

## Contents

## See also

- Weisstein, Eric W., Down Arrow Notation, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/DownArrowNotation.html]

#### Hierarchical list of operations pertaining to numbers ^{[1]} ^{[2]}

##### 0^{th} iteration

- Successor:

.S( *n*) - Predecessor:

.P( *n*)

##### 1^{st} iteration

- Addition:

, theS(S(⋯ " *a*times" ⋯ (S(*n*))))*sum*

, where*n*+*a*

is the*n**augend*and

is the*a**addend*. (When addition is commutative both are simply called*terms*.) - Subtraction:

, theP(P(⋯ " *s*times" ⋯ (P(*n*))))*difference*

, where*n*−*s*

is the*n**minuend*and

is the*s**subtrahend*.

##### 2^{nd} iteration

- Multiplication:

, the*n*+ (*n*+ (⋯ "*k*times" ⋯ (*n*+ (*n*))))*product*

, where*m*⋅*k*

is the*m**multiplicand*and

is the*k**multiplier*.^{[3]}(When multiplication is commutative both are simply called*factors*.) - Division: the
*ratio*

, where*n*/*d*

is the*n**dividend*and

is the*d**divisor*.- Quotient: (integer division).
- Remainder: (modulo and congruences).

##### 3^{rd} iteration

- Exponentiation (

as "degree",*d*

as "base",*b*

as "variable").*n*- Powers:

, written*n*⋅ (*n*⋅ (⋯ "*d*times" ⋯ (*n*⋅ (*n*))))

.*n**d* - Exponentials:

, written*b*⋅ (*b*⋅ (⋯ "*n*times" ⋯ (*b*⋅ (*b*))))

.*b**n*- Exponential function:

, where*e**n*

is Euler's number.*e*

- Exponential function:

- Powers:
- Exponentiation inverses (

as "degree",*d*

as "base",*b*

as "variable").*n*- Roots:

.*d*√*n* - Logarithms:

.log *b**n*- Natural logarithm function:

, orlog *n*

, wherelog *e**n*

is Euler's number.*e*

- Natural logarithm function:

- Roots:

##### 4^{th} iteration

- Tetration (

as "degree",*d*

as "base",*b*

as "variable").*n*- Tetra-powers (super-powers):

, written*n*^ (*n*^ (⋯ "*d*times" ⋯ (*n*^ (*n*))))

.*n*^^*d*or*n*↑↑*d* - Tetra-exponentials (super-exponentials):

, written*b*^ (*b*^ (⋯ "*n*times" ⋯ (*b*^ (*b*))))

.*b*^^*n*or*b*↑↑*n*

- Tetra-powers (super-powers):
- Tetration inverses (

as "degree",*d*

as "base",*b*

as "variable").*n*- Tetra-roots (super-roots)
- Tetra-logarithms (super-logarithms):

.slog *b**n*- Iterated logarithm:

.log ⁎ *b**n*= ⌈slog*b**n*⌉

- Iterated logarithm:

##### 5^{th} iteration

- Pentation (

as "degree",*d*

as "base",*b*

as "variable").*n*- Penta-powers:

, written*n*^^ (*n*^^ (⋯ "*d*times" ⋯ (*n*^^ (*n*^^ (*n*)))))

.*n*^^^*d*or*n*↑↑↑*d* - Penta-exponentials:

, written*b*^^ (*b*^^ (⋯ "*n*times" ⋯ (*b*^^ (*b*^^ (*b*)))))

.*b*^^^*n*or*b*↑↑↑*n*

- Penta-powers:
- Pentation inverses

##### 6^{th} iteration

- Hexation (

as "degree",*d*

as "base",*b*

as "variable").*n*- Hexa-powers:

, written*n*^^^ (*n*^^^ (⋯ "*d*times" ⋯ (*n*^^^ (*n*))))

.*n*^^^^*d*or*n*↑↑↑↑*d* - Hexa-exponentials:

, written*b*^^^ (*b*^^^ (⋯ "*n*times" ⋯ (*b*^^^ (*b*))))

.*b*^^^^*n*or*b*↑↑↑↑*n*

- Hexa-powers:
- Hexation inverses

##### 7^{th} iteration

- Heptation (

as "degree",*d*

as "base",*b*

as "variable").*n*- Hepta-powers:

, written*n*^^^^ (*n*^^^^ (⋯ "*d*times" ⋯ (*n*^^^^ (*n*))))

.*n*^^^^^*d*or*n*↑↑↑↑↑*d* - Hepta-exponentials:

, written*b*^^^^ (*b*^^^^ (⋯ "*n*times" ⋯ (*b*^^^^ (*b*))))

.*b*^^^^^*n*or*b*↑↑↑↑↑*n*

- Hepta-powers:
- Heptation inverses

##### 8^{th} iteration

- Octation (

as "degree",*d*

as "base",*b*

as "variable").*n*- Octa-powers:

, written*n*^^^^^ (*n*^^^^^ (⋯ "*d*times" ⋯ (*n*^^^^^ (*n*))))

.*n*^^^^^^*d*or*n*↑↑↑↑↑↑*d* - Octa-exponentials:

, written*b*^^^^^ (*b*^^^^^ (⋯ "*n*times" ⋯ (*b*^^^^^ (*b*))))

.*b*^^^^^^*n*or*b*↑↑↑↑↑↑*n*

- Octa-powers:
- Octation inverses

## Notes

- ↑ 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:

.*ω*× 2**:**=*ω*+*ω*