Hyper-Log notation or Hyper-L notation is an extension of the Log function that itself is a fast growing function, but not fast enough.[1] Hyper-L notation is as follows. The standard notation is L(n) to indicate it is a hyper-L function.
First Order[]
Representation: L(n)
Rule: \(L(n)\) = \(Antilog(n)\), also known as \(10^n\)
Examples:
- \(L(5)\) = 100,000
- \(L(33)\) = decillion
- \(L(100)\) = googol
- \(L(googol)\) = googolplex
Second Order[]
Representation: \(L((n))\)
Rule: \(L((n))\) = \(10^{Antilog(n)^{Antilog(n)^{...^{Antilog(n)}}}}; where ...=Antilog(n)\), or \(10^{10^{n}\uparrow \uparrow 10^n}\)
Examples:
- L((3)) = \(10^{1000^{1000^{...^{1000}}}} (... = 1{,}000)\)
- L((5)) = \(10^{100,000^{100,000^{...^{100,000}}}} (... = 100{,}000)\)
- L((33)) = \(10^{\text{decillion}^{\text{decillion}^{...^{\text{decillion}}}}} (... = \text{decillion})\)
- L((100)) = \(10^{\text{googol}^{\text{googol}^{\text{googol}^{...^{\text{googol}}}}}} (... = \text{googol})\)
- L((googol)) = \(10^{\text{googolplex}^{\text{googolplex}^{...^{\text{googolplex}}}}} (... = \text{googolplex})\)
Third Order[]
Representation: L(((n)))
Rule: \(L(((n))) = 10\uparrow^{Antilog(n)}Antilog(n)\)
Examples:
- L(((5))) = \(10\underbrace{\uparrow\uparrow\cdots\uparrow\uparrow}_{100{,}000}100{,}000\)
- L(((33))) = \(10\underbrace{\uparrow\uparrow\cdots\uparrow\uparrow}_{\text{decillion}}\text{decillion}\)
- L(((100))) = \(10\underbrace{\uparrow\uparrow\cdots\uparrow\uparrow}_{\text{googol}}\text{googol}\)
- L(((googol))) = \(10\underbrace{\uparrow\uparrow\cdots\uparrow\uparrow}_{\text{googolplex}}\text{googolplex}\)
Approximations[]
For first order (\(L(n)\)):
Notation | Approximation |
---|---|
Arrow notation | \(10\uparrow n\) (exact) |
Bowers' Exploding Array Function | {10,n} (exact) |
Chained arrow notation | \(10\longrightarrow n\) (exact) |
Copy notation | 9[n] |
Hyper-E notation | En (exact) |
Scientific notation | \(10^{n}\) (exact) |
X-Sequence Hyper-Exponential Notation | 10{1}n (exact) |
Fast-growing hierarchy | \(f_2(n)\) |
For second order (\(L((n))\)):
Notation | Approximation |
---|---|
Arrow notation | \(10\uparrow (10\uparrow n)\uparrow \uparrow (10\uparrow n)\) (exact) |
Bowers' Exploding Array Function | {10,{{10,n},{10,n},2}} (exact) |
Chained arrow notation | \(10\longrightarrow ((10\longrightarrow n)\longrightarrow (10\longrightarrow n)\longrightarrow 2)\) (exact) |
Hyper-E notation | E1#(En#En) |
Fast-growing hierarchy | \(f_{3}(n)\) |
For third order (\(L(((n)))\)):
Notation | Approximation |
---|---|
Bowers' Exploding Array Function | {10,{10,n},{10,n}} (exact) |
Chained arrow notation | \(10\longrightarrow (10\longrightarrow n)\longrightarrow (10\longrightarrow n)\) (exact) |
Hyper-E notation | E1##En |
Fast-growing hierarchy | \(f_{\omega}(f_1^3(n))\) |