Which is larger, e^pi or pi^e?

Which is greater, ${x^y}$ or ${y^x}$ ? Of course, it depends on the values of x and y. We might consider a particular case: Is ${e^\pi > \pi^e}$ or ${\pi^e > e^\pi}$ ?

x2yminusy2x — Contour plot of x^y – y^x, positive in the yellow regions, negative in the blue ones.

We assume that ${x}$ and ${y}$ are positive real numbers, and plot the function

$\displaystyle z(x,y) = x^y - y^x$

In the contour plot, we show the areas of the quadrant ${x > 0}$ , ${y > 0}$ where the function ${z(x,y)}$ is positive (shaded yellow) and where it is negative (shaded blue). The quadrant is divided into four regions by two curves upon which ${z=0}$ . The first is the diagonal line ${x = y}$ , which clearly makes ${z}$ vanish. The second is a curve that passes through the point ${(x,y) = (e,e)}$ . This hyperbola-like curve is asymptotic to the lines ${x=1}$ and ${y=1}$ .

What is the nature of this curve? Since it corresponds to ${x^y=y^x}$ , we have

$\displaystyle x^y=y^x \quad \Longrightarrow\quad y\log x = x\log y \quad \Longrightarrow\quad \frac{\log x}{x} = \frac{\log y}{y}$

Obviously, ${x=y}$ solves this. But there are other solutions. Drawing the graph of ${\xi(x) = \log x / x}$ we see that, for any ${x > 1}$ there is another point ${y > 1}$ for which ${\xi(y)=\xi(x)}$ .

The function ${\xi}$ 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 ${e^\pi > \pi^e}$ or ${\pi^e > e^\pi}$ ? We locate the point ${(e,\pi)}$ in the quadrant. If it falls in a yellow region, then ${e^\pi > \pi^e}$ . If in a blue region then ${e^\pi < \pi^e}$ .

Clearly, ${(e,\pi)}$ is in the yellow region directly above the point ${(e, e)}$ . Therefore ${e^\pi > \pi^e}$ . A calculator confirms that ${e^\pi=23.14}$ and ${\pi^e = 22.46}$ .