Graphical Comparison of Exponential Functions

This example shows an interesting graphical approach for discovering whether $e^{π}$ is greater than $π^{e}$ .

The question is: which is greater, $e^{π}$ or $π^{e}$ ? The easy way to find out is to type it directly at the MATLAB® command prompt. But another way to analyze the situation is to ask a more general question: what is the shape of the function $z (x, y) = x^{y} - y^{x}$ ?

Here is a plot of $z$ .

% Define the mesh
x = 0:0.16:5;
y = 0:0.16:5;
[xx,yy] = meshgrid(x,y);

% The plot
zz = xx.^yy-yy.^xx;
h = surf(x,y,zz);
h.EdgeColor = [0.7 0.7 0.7];
view(20,50);
colormap(hsv);
title('$z = x^y-y^x$','Interpreter','latex')
xlabel('x')
ylabel('y')
hold on

Figure contains an axes object. The axes object with title z equals x toThePowerOf y baseline minus y toThePowerOf x baseline, xlabel x, ylabel y contains an object of type surface.

The solution of the equation $x^{y} - y^{x} = 0$ has a very interesting shape, and our original question is not easily solved by inspection. Here is a plot of the xy values that yield $z = 0$ .

c = contourc(x,y,zz,[0 0]);
list1Len = c(2,1);
xContour = [c(1,2:1+list1Len) NaN c(1,3+list1Len:size(c,2))];
yContour = [c(2,2:1+list1Len) NaN c(2,3+list1Len:size(c,2))];
% Note that the NAN above prevents the end of the first contour line from being
% connected to the beginning of the second line
line(xContour,yContour,'Color','k');

Figure contains an axes object. The axes object with title z equals x toThePowerOf y baseline minus y toThePowerOf x baseline, xlabel x, ylabel y contains 2 objects of type surface, line.

Some combinations of x and y along the black curve are integers. This next plot is of the integer solutions to the equation $x^{y} - y^{x} = 0$ . Notice that $2^{4} = 4^{2}$ is the only integer solution where $x \neq y$ .

plot([0:5 2 4],[0:5 4 2],'r.','MarkerSize',25);

Figure contains an axes object. The axes object with title z equals x toThePowerOf y baseline minus y toThePowerOf x baseline, xlabel x, ylabel y contains 3 objects of type surface, line. One or more of the lines displays its values using only markers

Finally, plot the points $(π, e)$ and $(e, π)$ on the surface. The result shows that $e^{π}$ is indeed larger than $π^{e}$ (though not by much).

e = exp(1);
plot([e pi],[pi e],'r.','MarkerSize',25);
plot([e pi],[pi e],'y.','MarkerSize',10);
text(e,3.3,'(e,pi)','Color','k', ...
   'HorizontalAlignment','left','VerticalAlignment','bottom');
text(3.3,e,'(pi,e)','Color','k','HorizontalAlignment','left',...
   'VerticalAlignment','bottom');
hold off;

Figure contains an axes object. The axes object with title z equals x toThePowerOf y baseline minus y toThePowerOf x baseline, xlabel x, ylabel y contains 7 objects of type surface, line, text. One or more of the lines displays its values using only markers

Verify the results.

e = exp(1);
e^pi

ans = 
23.1407

pi^e

ans = 
22.4592

Graphical Comparison of Exponential Functions

See Also