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. 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 $$10\uparrow n$$