Iterated Logarithm log*(n)

Iterated Logarithm or Log*(n) is the number of times the logarithm function must be iteratively applied before the result is less than or equal to 1.

   \log ^{*}n:={\begin{cases}0}n\leq 1;\\1+\log ^{*}(\log n)}n>1\end{cases}}}

Applications: It is used in analysis of algorithms (Refer Wiki for details)

C++

// Recursive CPP program to find value of
// Iterated Logarithm
#include <bits/stdc++.h>
using namespace std;
  
int _log(double x, double base)
{
    return (int)(log(x) / log(base));
}
  
double recursiveLogStar(double n, double b)
{
    if (n > 1.0)
        return 1.0 + recursiveLogStar(_log(n, b), b);
    else
        return 0;
}
  
// Driver code
int main()
{
    int n = 100, base = 5;
    cout << "Log*(" << n << ") = " 
         << recursiveLogStar(n, base) << "\n";
    return 0;
}

Java

// Recursive Java program to 
// find value of Iterated Logarithm
import java.io.*;
  
class GFG
{
static int _log(double x, 
                double base)
{
    return (int)(Math.log(x) / 
                 Math.log(base));
}
  
static double recursiveLogStar(double n, 
                               double b)
{
    if (n > 1.0)
        return 1.0
               recursiveLogStar(_log(n, 
                                 b), b);
    else
        return 0;
}
  
// Driver code
public static void main (String[] args) 
{
    int n = 100, base = 5;
    System.out.println("Log*(" + n + ") = " +
                  recursiveLogStar(n, base));
}
}
  
// This code is contributed by jit_t

Python3

# Recursive Python3 program to find value of
# Iterated Logarithm
import math
  
def _log(x, base):
  
    return (int)(math.log(x) / math.log(base))
  
def recursiveLogStar(n, b):
  
    if(n > 1.0):
        return 1.0 + recursiveLogStar(_log(n, b), b)
    else:
        return 0
  
  
# Driver code
if __name__=='__main__':
    n = 100
    base = 5
    print("Log*(", n, ") = ", recursiveLogStar(n, base))
  
# This code is contributed by
# Sanjit_Prasad

PHP

<?php
// Recursive PhP program to find 
// value of Iterated Logarithm
  
function _log($x, $base)
{
    return (int)(log($x) / log($base));
}
  
function recursiveLogStar($n, $b)
{
    if ($n > 1.0)
        return 1.0 + 
               recursiveLogStar(_log($n
                               $b), $b);
    else
        return 0;
}
  
// Driver code
$n = 100; $base = 5;
echo "Log*(" , $n , ")"," = ",
recursiveLogStar($n, $base), "\n";
  
// This code is contributed by ajit
?>


Output :

Log*(100) = 2

Iterative Implementation :

C++

// Iterative CPP function to find value of
// Iterated Logarithm
int iterativeLogStar(double n, double b)
{
    int count = 0;
    while (n >= 1) {
        n = _log(n, b);
        count++;
    }
    return count;
}

Python3

# Iterative Python function to find value of
# Iterated Logarithm
  
  
def iterativeLogStar(n, b):
  
    count = 0
    while(n >= 1):
        n = _log(n, b)
        count = count + 1
  
    return count
  
# This code is contributed by
# Sanjit_Prasad

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Improved By : jit_t, Sanjit_Prasad



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