Calculus

Which Number is Larger?

This problem is quite famous, and you might have seen a few special cases before, but in this article, we will solve the generalized version so that you will be able to tell what the answer is in just a matter of seconds.

Solution

One natural thing that comes to mind is to just take the logarithm of both sides. This will make it easier to deal with the exponents. Then, we can do some rearranging:

Now we can define a function and it’s only left to analyze it

Thus it reaches its extrema at x = e. Now if we plug in some number less than e, we get that the derivative is positive (which means that f is increasing) and if we plug in a number that is greater than e, then the derivative will be negative (so, the function will be decreasing). Thus we get the following table:

And here is the actual graph of the function:

So, returning to our original problem, we conclude that

• If y < x ≤ e, then x^y > y^x
• If e < y < x, then y^x > x^y

Conclusion

Hopefully, you found this article interesting and useful. Now you will be able to compare some numbers of the form x^y and y^x much more easily, without doing any boring arithmetic.