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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:

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 \(f_{\omega}(f_1^3(n))\)

Sources[]

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