Hyper-L Notation

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)= 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 ... equals revlog(n)



Examples:

L((3)) = 10^1000^1000^...^1000 (... = 1,000)

L((5))= 10^100,000^100,000^...^100,000 (... = 100,000)

<p style="font-weight:normal;font-size:16px;">L((33))= 10^decillion^decillion^...^decillion (... = decillion)

<p style="font-weight:normal;font-size:16px;">L((100))= 10^googol^googol^googol^...^googol (... = googol)

<p style="font-weight:normal;font-size:16px;">L((googol))= 10^googolplex^googolplex^...^googolplex (... = googolplex)

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Third Order
<p style="font-weight:normal;margin-top:1em;margin-bottom:1em;font-size:16px;">Representation: L(((n)))

<p style="font-weight:normal;font-size:16px;">Rule: <span style="line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;"> <span style="vertical-align:0px;display:inline-block;line-height:normal;width:6.513em;height:0px;font-size:19.2px;"> <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">10 <span style="padding-left:0.263em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">↑ ↑  <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">. <span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">. <span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">. <span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">↑ ↑ <span style="padding-left:0.263em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">( <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">N <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">)      with revlog(n) up arrows

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<p style="font-weight:normal;font-size:16px;">Examples:

<p style="font-weight:normal;font-size:16px;">L(((5)))= <span style="line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;"> <span style="vertical-align:0px;display:inline-block;line-height:normal;width:9.013em;height:0px;font-size:19.2px;"> <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">10 <span style="padding-left:0.263em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">↑ ↑  <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">. <span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">. <span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">. <span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">↑ ↑ <span style="padding-left:0.263em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">( <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">100 <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">, <span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">000 <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">)      with 100,000 up arrows

<p style="font-weight:normal;font-size:16px;">L(((33)))= <span style="line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;"> <span style="vertical-align:0px;display:inline-block;line-height:normal;width:9.378em;height:0px;font-size:19.2px;"> <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">10 <span style="padding-left:0.263em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">↑ ↑  <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">. <span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">. <span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">. <span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">↑ ↑ <span style="padding-left:0.263em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">( <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">d <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">e <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">c <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">i <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">l <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">l <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">i <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">o <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">n <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">)      with decillion up arrows

<p style="font-weight:normal;font-size:16px;">L(((100)))= <span style="line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;"> <span style="vertical-align:0px;display:inline-block;line-height:normal;width:8.336em;height:0px;font-size:19.2px;"> <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">10 <span style="padding-left:0.263em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">↑ ↑  <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">. <span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">. <span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">. <span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">↑ ↑ <span style="padding-left:0.263em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">( <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">g <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">o <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">o <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">g <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">o <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">l <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">)       with googol up arrows

<p style="font-weight:normal;font-size:16px;">L(((googol)))= <span style="line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;"> <span style="vertical-align:0px;display:inline-block;line-height:normal;width:10.159em;height:0px;font-size:19.2px;"> <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">10 <span style="padding-left:0.263em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">↑ ↑  <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">. <span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">. <span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">. <span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">↑ ↑ <span style="padding-left:0.263em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">( <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">g <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">o <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">o <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">g <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">o <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">l <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">p <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">l <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">e <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">x <span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">)      with googolplex up arrows

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