# 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.

**Applications:** It is used in the 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` |

## C#

`// Recursive C# program to ` `// find value of Iterated Logarithm` `using` `System;` `public` `class` `GFG{` `static` `int` `_log(` `double` `x, ` `double` `baset)` `{` ` ` `return` `(` `int` `)(Math.Log(x) / ` ` ` `Math.Log(baset));` `}` `static` `double` `recursiveLogStar(` `double` `n, ` ` ` `double` `b)` `{` ` ` `if` `(n > 1.0)` ` ` `return` `1.0 + ` ` ` `recursiveLogStar(_log(n, ` ` ` `b), b);` ` ` `else` ` ` `return` `0;` `}` `// Driver code` ` ` `static` `public` `void` `Main (){` ` ` ` ` `int` `n = 100, baset = 5;` ` ` `Console.WriteLine(` `"Log*("` `+ n + ` `") = "` `+` ` ` `recursiveLogStar(n, baset));` `}` `}` `// This code is contributed by ajit.` |

## 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` `?>` |

## Javascript

`<script>` `// Javascript program to` `// find value of Iterated Logarithm` ` ` `function` `_log( x, base)` `{` ` ` `return` `(Math.log(x) /` ` ` `Math.log(base));` `}` ` ` `function` `recursiveLogStar(n, b)` `{` ` ` `if` `(n > 1.0)` ` ` `return` `1.0 +` ` ` `recursiveLogStar(_log(n,` ` ` `b), b);` ` ` `else` ` ` `return` `0;` `}` ` ` `// Driver code` ` ` `let n = 100, base = 5;` ` ` `document.write(` `"Log*("` `+ n + ` `") = "` `+` ` ` `recursiveLogStar(n, base));` ` ` ` ` `// This code is contributed by sanjoy_62.` `</script>` |

**Output :**

Log*(100) = 2

**Time Complexity: **O(logn)

**Auxiliary Space:** O(logn) due to recursive stack space**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;` `}` |

## Java

`// Iterative Java function to find value of` `// Iterated Logarithm` `public` `static` `int` `iterativeLogStar(` `double` `n, ` `double` `b)` `{` ` ` `int` `count = ` `0` `;` ` ` `while` `(n >= ` `1` `) {` ` ` `n = _log(n, b);` ` ` `count++;` ` ` `}` ` ` `return` `count;` `}` `// This code is contributed by pratham76` |

## 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` |

## C#

`// Iterative C# function to find value of` `// Iterated Logarithm` `static` `int` `iterativeLogStar(` `double` `n, ` `double` `b)` `{` ` ` `int` `count = 0;` ` ` `while` `(n >= 1)` ` ` `{` ` ` `n = _log(n, b);` ` ` `count++;` ` ` `}` ` ` `return` `count;` `}` `// This code is contributed by rutvik_56` |

## Javascript

`<script>` `// Iterative javascript function to find ` `// value of Iterated Logarithm` `function` `iterativeLogStar(n, b)` `{` ` ` `var` `count = 0;` ` ` `while` `(n >= 1)` ` ` `{` ` ` `n = _log(n, b);` ` ` `count++;` ` ` `}` ` ` `return` `count;` `}` `// This code is contributed by 29AjayKumar ` `</script>` |

**Time Complexity: **O(logn)

**Auxiliary Space:** O(1)

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