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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: \(L((n))\) = \(10^{Antilog(n)^{Antilog(n)^{...^{Antilog(n)}}}}; where ...=Antilog(n)\)

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 Lower bound Upper bound
Hyper-E notation E(n)
Arrow notation \(10\uparrow n\)

Sources

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