Which is greater, or ? Of course, it depends on the values of x and y. We might consider a particular case: Is or ?
We assume that and are positive real numbers, and plot the function
In the contour plot, we show the areas of the quadrant , where the function is positive (shaded yellow) and where it is negative (shaded blue). The quadrant is divided into four regions by two curves upon which . The first is the diagonal line , which clearly makes vanish. The second is a curve that passes through the point . This hyperbola-like curve is asymptotic to the lines and .
What is the nature of this curve? Since it corresponds to , we have
Obviously, solves this. But there are other solutions. Drawing the graph of we see that, for any there is another point for which .
The function is related to the Lambert W-function. Analytically-minded readers might like to establish this relationship explicitly.
Is e^pi less than or greater than pi^e?
We return now to the simple question: is or ? We locate the point in the quadrant. If it falls in a yellow region, then . If in a blue region then .
Clearly, is in the yellow region directly above the point . Therefore . A calculator confirms that and .