Nth root of a number using log

Last Updated : 18 Sep, 2022

Given two integers N and K, the task is to find the N^th root of the K.

Examples:

Input: N = 3, K = 8
Output: 2.00
Explanation:
Cube root of 8 is 2. i.e. 2³ = 8

Input: N = 2, K = 16
Output: 4.00
Explanation:
Square root of 16 is 4, i.e. 4² = 16

Approach: The idea is to use logarithmic function to find the N^th root of K.

Let D be our N^th root of the K,
Then, N^{\frac{1}{K}} = D
Apply log_K on both the sides -
=> log_{K}(N^{\frac{1}{K}}) = log_{K}(D)
=> \frac{1}{K} * log_{K}(N) = log_{K}(D)
=> D = K^{\frac{1}{K} * log_{K}(N)}

Below is the implementation of the above approach:

C++

// C++ implementation to find the
// Kth root of a number using log

#include <bits/stdc++.h>

// Function to find the Kth root
// of the number using log function
double kthRoot(double n, int k)
{
    return pow(k,
               (1.0 / k)
                   * (log(n)
                      / log(k)));
}

// Driver Code
int main(void)
{
    double n = 81;
    int k = 4;
    printf("%lf ", kthRoot(n, k));
    return 0;
}

Java

// Java implementation to find the
// Kth root of a number using log
import java.util.*;

class GFG {

// Function to find the Kth root
// of the number using log function
static double kthRoot(double n, int k)
{
    return Math.pow(k, ((1.0 / k) * 
                       (Math.log(n) /
                        Math.log(k))));
}

// Driver Code
public static void main(String args[]) 
{
    double n = 81;
    int k = 4;
    
    System.out.printf("%.6f", kthRoot(n, k));
}
}

// This code is contributed by rutvik_56

Python3

# Python3 implementation to find the 
# Kth root of a number using log

import numpy as np 

# Function to find the Kth root 
# of the number using log function 
def kthRoot(n, k): 
    
    return pow(k, ((1.0 / k) * 
                  (np.log(n) / 
                   np.log(k))))
                   
# Driver Code 
n = 81
k = 4

print("%.6f" % kthRoot(n, k))

# This code is contributed by PratikBasu

// C# implementation to find the
// Kth root of a number using log
using System;

class GFG {

// Function to find the Kth root
// of the number using log function
static double kthRoot(double n, int k)
{
    
    return Math.Pow(k, ((1.0 / k) * 
                        (Math.Log(n) /
                         Math.Log(k))));
}

// Driver Code
public static void Main(String []args) 
{
    double n = 81;
    int k = 4;
    
    Console.Write("{0:F6}", kthRoot(n, k));
}
}

// This code is contributed by AbhiThakur

JavaScript

<script>

// Javascript implementation to find the
// Kth root of a number using log

// Function to find the Kth root
// of the number using log function
function kthRoot(n, k)
{
   return Math.pow(k, ((1.0 / k) * 
                       (Math.log(n) /
                        Math.log(k))));
}
 
// Driver Code
var n = 81;
var k = 4;
var x = kthRoot(n, k)

document.write(x.toFixed(6));

// This code is contributed by Ankita saini
                    
</script>

Output:

3.000000

Time Complexity: O(1)

Auxiliary Space: O(1)

Nth root of a number using log

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Nth root of a number using log

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