Manuals/calci/LOGSTAR

LOGSTAR (Number,Base)

• $Number$ is any real number.
• $Base$ base value of the log.

Description

• This function shows the iterated logarithm of the given number.
• In $LOGSTAR(Number,Base)$, $Number$ is any real number.
• $Base$ is the base value of the log.
• Log star is the number of times the logarithm function must be iteratively applied before the result is less than or equal to 1.
• The simple definition of the Log star is :

${\begin{cases}0&ifn\leq 1\\1+log*(logn)&ifn>1\\\end{cases}}$

• The relation between Log star and Super logarithm is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle log*n=[slog_e(n)]} for positive real numbers.
• Also for negative real numbers ,log star value is 0,whereas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [slog_e(-x)]= -1} for positive x.
• The iterated logarithm varies at slow rate,which is very slow compared with logarithm values.
• Iterated logarithm values vary not more than 5 with base value 2.

Spreadsheet

X	Log*X
(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\infty} ,1]	0
(1,2]	1
(2,4]	2
(4,16]	3
(16, 65536]	4
(65536, 2^65536]	5