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}.


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