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Graphical Comparison of Exponential Functions

This example shows an interesting graphical approach for discovering whether eπ is greater than πe.

The question is: which is greater, eπ or πe? 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)=xy-yx?

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 contains an object of type surface.

The solution of the equation xy-yx=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 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 xy-yx=0. Notice that 24=42 is the only integer solution where xy.

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 contains 3 objects of type surface, line.

Finally, plot the points (π,e) and (e,π) on the surface. The result shows that eπ is indeed larger than πe (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 contains 7 objects of type surface, line, text.

Verify the results.

e = exp(1);
e^pi
ans = 23.1407
pi^e
ans = 22.4592

See Also

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