Encuentre el valor de ln(N!) usando Recursion

Dado un número N, la tarea es encontrar el valor logarítmico del factorial de N, es decir, log(N!).
Nota: ln significa log con base e.
Ejemplos: 
 

Input: N = 2
Output: 0.693147

Input:  N = 3
Output: 1.791759

Enfoque:
Método -1: Calcular n! primero, luego tome su valor de registro.
Método -2: Utilizando la propiedad de log, es decir, tomar la suma de los valores de log de n, n-1, n-2…1. 
 

ln(n!) = ln(n*n-1*n-2*…..*2*1) = ln(n)+ln(n-1)+……+ln(2)+ln(1 ) 
 

A continuación se muestra la implementación del Método-2: 
 

// C++ implementation of the above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to calculate the value
double fact(int n)
{
    if (n == 1)
        return 0;
    return fact(n - 1) + log(n);
}
// Driver code
int main()
{
    int N = 3;
    cout << fact(N) << "\n";
    return 0;
}

// C implementation of the above approach
#include <math.h>
#include <stdio.h>
 
long double fact(int n)
{
    if (n == 1)
        return 0;
    return fact(n - 1) + log(n);
}
 
// Driver code
int main()
{
    int n = 3;
    printf("%Lf", fact(n));
    return 0;
}

// Java implementation of the above approach
import java.util.*;
import java.io.*;
class logfact {
    public static double fact(int n)
    {
        if (n == 1)
            return 0;
        return fact(n - 1) + Math.log(n);
    }
 
    public static void main(String[] args)
    {
 
        int N = 3;
        System.out.println(fact(N));
    }
}

# Python implementation of the above approach
import math
def fact(n):
    if (n == 1):
        return 0
    else:
        return fact(n-1) + math.log(n)
N = 3
print(fact(N))

// C# implementation of the above approach
using System;
 
class GFG
{
    public static double fact(int n)
    {
        if (n == 1)
            return 0;
        return fact(n - 1) + Math.Log(n);
    }
 
    // Driver code
    public static void Main()
    {
        int N = 3;
        Console.WriteLine(fact(N));
    }
}
 
// This code is contributed by ihritik

<?php
 
//PHP implementation of the above approach
 
function fact($n)
{
    if ($n == 1)
        return 0;
    return fact($n - 1) + log($n);
}
 
// Driver code
$n = 3;
echo fact($n);
 
// This code is contributed by ihritik
?>

<script>
 
// Javascript implementation of the above approach
 
// Function to calculate the value
function fact(n)
{
    if (n == 1)
        return 0;
    return fact(n - 1) + Math.log(n);
}
// Driver code
var N = 3;
document.write( fact(N).toFixed(6) + "<br>");
 
</script>

1.791759

Complejidad de tiempo: O(n)

Espacio Auxiliar: O(n)