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The "penta-logarithm"
![{\displaystyle b\downarrow \downarrow \downarrow n\equiv \log _{b}^{**}(n)=d,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/fbfce93e39c4ae860b6783740e0b074c91e71036)
is the logarithmic-like inverse of the "penta-exponential"
![{\displaystyle b\uparrow \uparrow \uparrow d\equiv \exp _{b}^{**}(d)=n.\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/f55897e540abedf23afee180f2f20081619854d0)
The ceiling of the "penta-logarithm" of a positive integer
denotes the number of iterations of the tetra-logarithm (base
) that is required such that
![{\displaystyle \underbrace {\log _{b}^{*}\ldots \log _{b}^{*}} _{\lceil d\rceil }n\leq b,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/9bcfc1716a1457eaa22812b75f19ebaebfa34904)
where
is a nonnegative integer when
is of the form
![{\displaystyle n=\underbrace {\exp _{b}^{*}\ldots \exp _{b}^{*}} _{d}1,\quad d\geq 0.\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/58a7b48dfc65ec8281ba2e530f703487f0faf101)
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 "penta-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 , 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.
2nd iteration
- Multiplication:
n + (n + (⋯ "k times" ⋯ (n + (n)))) |
, the product , where is the multiplicand and is the multiplier.[3] (When multiplication is commutative both are simply called factors.)
- Division: the ratio , where is the dividend and is the divisor.
3rd 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").
4th iteration
- Tetration ( as "degree", as "base", as "variable").
- Tetration inverses ( as "degree", as "base", as "variable").
5th iteration
- Pentation ( as "degree", as "base", as "variable").
- Pentation inverses
6th iteration
- Hexation ( as "degree", as "base", as "variable").
- Hexation inverses
7th iteration
- Heptation ( as "degree", as "base", as "variable").
- Heptation inverses
8th 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
Notes