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: \(L(n)\) = reverse \(log(n)\) or \(revlog(n)\)
Examples:
- \(L(5)\) = 100,000
- \(L(33)\) = decillion
- \(L(100)\) = googol
- \(L(googol)\) = googolplex
Second Order
Representation: \(L((n))\)
Rule: \(L((n))\) = \(10^{revlog(n)^{revlog(n)^{...^{revlog(n)}}}}; where ...=revlog(n)\)
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: \(L(((n))) = 10\uparrow^{revlog(n)}\) with revlog(n) up arrows
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))) = \(10\underbrace{\uparrow\uparrow\cdots\uparrow\uparrow}_{text{googolplex}}text{googolplex}\)