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The "octa-logarithm"

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

The ceiling of the "octa-logarithm" of a positive integer denotes the number of iterations of the hepta-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 "octa-logarithm" yet, although the down-arrow notation (derived from Knuth's up-arrow notation) seems the most intuitive one.

See also

Hierarchical list of operations pertaining to numbers [1] [2]

0th iteration
1st iteration
  • Addition:
    S(S( "a times" (S(n))))
    , the sum
    n  +  a
    , 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
    n  −  s
    , where
    is the minuend and
    is the subtrahend.
2nd iteration
3rd iteration
4th iteration
5th iteration
6th iteration
7th iteration
8th iteration
  • Octation (
    as "degree",
    as "base",
    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


  2. Grzegorczyk
  3. 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