Graphical comparison of exponential functions

This example discusses a graphical approach to determining the larger of two values: $e^\pi$ and $\pi^e$.

To graphically compare the values of $e^\pi$ and $\pi^e$, we plot the surface of the difference function of the exponential functions $z=x^y-y^x$.

To do this, first of all, install and connect the library CairoMakie.

import Pkg; Pkg.add("CairoMakie");
using CairoMakie;

Let's define arrays of surface coordinates.

x = 0:0.05:4; y = 0:0.05:4;
X = [i for i in x, j in 1:length(y)]; Y = [j for i in 1:length(x), j in y];
Z = X.^Y-Y.^X;

Build the surface by the given function.

using CairoMakie

# Создаем фигуру и 3D оси
fig = Figure(;size = (600, 600));
ax = Axis3(fig[1, 1], 
          title = "Показательные функции",
          xlabel = "Ось X", ylabel = "Ось Y", zlabel = "Ось Z",
          aspect=(1,1,1), azimuth=-pi*0.64);

# Создаём поверхность
CairoMakie.surface!(ax, x, y, Z, 
                    colormap = :prism,
                    colorrange = (minimum(Z), maximum(Z)));

# Добавляем цветовую шкалу
Colorbar(fig[1, 2],
         limits = (minimum(Z), maximum(Z)),
         colormap = :prism, label = "Значения");

display(fig);

In the plane $Z=0$ draw the contour on the constructed surface.

CairoMakie.contour3d!(ax, x, y, Z,
                      levels = [0],
                      color=:black, linewidth=2)
display(fig);

On the constructed contours there are several points of integer solutions of the equation $x^y-y^x=0$. Let us plot these points on the contours.

# массивы целочисленных решений уравнения
ix=[0,1,2,2,3,4,4] 
iy=[0,1,2,4,3,2,4]
iz=zeros(7)

CairoMakie.scatter!(ax, ix, iy, iz;
                    color=:black)
display(fig);

Now plot the function points for the sought $x$ and $y$ at the coordinates: $(\pi,\, e,\, \pi^e-e^\pi)$ and $(e,\, \pi,\, e^\pi-\pi^e)$.

e=exp(1);
tx=[pi,e]
ty=[e,pi]
textt=["pi,e", "e,pi"]
tz=tx.^ty-ty.^tx

CairoMakie.scatter!(ax, tx, ty, tz;
                    color=:white)
CairoMakie.text!(ax, tx, ty, tz;
                 text = textt, align = (:right, :bottom),
                 color = :white)
display(fig);

From the construction we can see that the point with coordinates $(e,\, \pi,\, e^\pi-\pi^e)$ is located in the part of the surface that is above the plane $Z = 0$, and the point $(\pi,\, e,\, \pi^e-e^\pi)$ - in the part of the surface below the secant plane. For convenience of viewing the location of points you can edit the azimuth of axes and code cell mask:

Азимут=-0.38 # @param {type:"slider",min:-6.28,max:6.28,step:0.1}
ax.azimuth[] = Азимут;
display(fig);

From this we can conclude that the value of $e^\pi$ is greater than $\pi^e$. In confirmation let's calculate these values.

@show e^pi, pi^e;

(e ^ pi, pi ^ e) = (23.140692632779263, 22.459157718361038)

Conclusion

In this example, we have looked at an illustrative way to graphically find the larger of two values.