Comparing X^Y and Y^X for very large values of X and Y

Namachila November 19, 2025

0 0 3 minutes read

Given two integer X and Y, the task is compare X^Y and Y^X for large values of X and Y.
Examples:

Input: X = 2, Y = 3
Output: 2^3 < 3^2
2³ < 3²
Input: X = 4, Y = 5
Output: 4^5 > 5^4

Naive approach: A basic approach is to find the values X^Y and Y^X and compare them which can overflow as the values of X and Y can be large
Better approach: Taking log of both the equations, log(X^Y) = Y * log(X) and log(Y^X) = X * log(Y). Now, these values can be compared easily without overflows.
Below is the implementation of the above approach:

C++

// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to compare x^y and y^x
void compareVal(int x, int y)
{
 
    // Storing values OF x^y AND y^x
    long double a = y * log(x);
    long double b = x * log(y);
 
    // Comparing values
    if (a > b)
        cout << x << "^" << y << " > "
             << y << "^" << x;
 
    else if (a < b)
        cout << x << "^" << y << " < "
             << y << "^" << x;
 
    else if (a == b)
        cout << x << "^" << y << " = "
             << y << "^" << x;
}
 
// Driver code
int main()
{
    long double x = 4, y = 5;
 
    compareVal(x, y);
 
    return 0;
}

Java

// Java implementation of the approach
import java.util.*;
 
class GFG
{
     
// Function to compare x^y and y^x
static void compareVal(int x, int y)
{
 
    // Storing values OF x^y AND y^x
    double a = y * Math.log(x);
    double b = x * Math.log(y);
 
    // Comparing values
    if (a > b)
        System.out.print(x + "^" + y + " > " + 
                         y + "^" + x);
 
    else if (a < b)
        System.out.print(x + "^" + y + " < " +
                         y + "^" + x);
 
    else if (a == b)
        System.out.print(x + "^" + y + " = " +
                         y + "^" + x );
}
 
// Driver code
public static void main(String[] args) 
{
    int x = 4, y = 5;
 
    compareVal(x, y);
}
} 
 
// This code is contributed by 29AjayKumar

Python3

# Python3 implementation of the approach 
from math import log
 
# Function to compare x^y and y^x 
def compareVal(x, y) :
     
    # Storing values OF x^y AND y^x
    a = y * log(x);
    b = x * log(y);
     
    # Comparing values
    if (a > b) :
        print(x, "^", y, ">", y, "^", x);
         
    elif (a < b) :
        print(x, "^", y, "<", y ,"^", x);
 
    elif (a == b) :
        print(x, "^", y, "=", y, "^", x); 
 
# Driver code 
if __name__ == "__main__" : 
 
    x = 4; y = 5; 
 
    compareVal(x, y); 
 
# This code is contributed by AnkitRai01

C#

// C# implementation of the approach
using System;
class GFG
{
     
// Function to compare x^y and y^x
static void compareVal(double x, double y)
{
 
    // Storing values OF x^y AND y^x
    double a = y * Math.Log(x);
    double b = x * Math.Log(y);
 
    // Comparing values
    if (a > b)
        Console.Write (x + "^" + y + " > " + 
                       y + "^" + x);
 
    else if (a < b)
            Console.Write (x + "^" + y + " < "+
                           y + "^" + x);
 
    else if (a == b)
        Console.Write (x + "^" + y + " = " +
                       y + "^" + x );
}
 
// Driver code
static public void Main ()
{
    double x = 4, y = 5;
 
    compareVal(x, y);
}
}
 
// This Code is contributed by ajit. 

Javascript

<script>
// Javascript implementation of the approach
 
// Function to compare x^y and y^x
function compareVal(x, y)
{
 
    // Storing values OF x^y AND y^x
    let a = y * Math.log(x);
    let b = x * Math.log(y);
 
    // Comparing values
    if (a > b)
        document.write(x + "^" + y + " > "
             + y + "^" + x);
 
    else if (a < b)
        document.write(x + "^" + y + " < "
             + y + "^" + x);
 
    else if (a == b)
        document.write(x + "^" + y + " = "
             + y + "^" + x);
}
 
// Driver code
    let x = 4, y = 5;
 
    compareVal(x, y);
 
</script>

Output:

4^5 > 5^4

Time Complexity: O(1)

Auxiliary Space: O(1)

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