Manuals/calci/LBSTAR

LBSTAR (Number)

• 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 Number} is any real number.

Description

• This function shows the iterated logarithm of the given number with the base 2.
• In 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 LBSTAR (Number)} , 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 Number} is any real number.
• 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 :

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 \begin{cases} 0 & if n\le 1 \\ 1+log*(log n) & ifn > 1 \\ \end{cases} }

• The relation between Log star and Super logarithm is $log*n=[slog_{e}(n)]$ 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, 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 2^{65536}} ]	5

Examples

1. LBSTAR(6) = 3
2. LBSTAR(300) = 4
3. LBSTAR(67230) = 5

See Also

• LOGSTAR
• LOG10
• LOGINV

References

• Log star

• List of Main Z Functions
• Z3 home