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Representation: \(L((n))\) |
Representation: \(L((n))\) |
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− | Rule: \(L((n))\) = \(10^{Antilog(n)^{Antilog(n)^{...^{Antilog(n)<nowiki>}}}</nowiki>}; where ...= |
+ | Rule: \(L((n))\) = \(10^{Antilog(n)^{Antilog(n)^{...^{Antilog(n)<nowiki>}}}</nowiki>}; where ...=Antilog(n)\) |
Examples: |
Examples: |
Revision as of 15:22, 8 December 2020
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: \(L((n))\) = \(10^{Antilog(n)^{Antilog(n)^{...^{Antilog(n)}}}}; where ...=Antilog(n)\)
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 fisrt order (\(L(n)\)):
Notation | Lower bound | Upper bound |
---|---|---|
Hyper-E notation | E(n) | |
Arrow notation | \(10\uparrow n\) |