## An interesting property of the number e

**NOTE**: Please WAIT while the applets below load. If a security pop-up menu appears in your browser, please choose **RUN/ALLOW** to let the applets run properly. *It is completely safe & can be trusted*. If you have the very latest Java on your PC or Apple Mac, and experience problems with the applets loading, please go *here* for additional information on Java settings that should resolve the issue.

The natural number e = 2.718...., the base of the natural logarithm, plays a fundamental role in calculus. It has many interesting properties.

Below are shown two graphs, namely, *a*^{x} and *x*^{a}. Note that the exponential function graph in blue for the initial value of *a* = 2.00 (to 2 decimals) lies above the power function graph in green, except between *x* = 2 and *x* = 4, where the exponential graph is below the power graph.

**Explore**: Is this always the case for any value of the base *a*, that the exponential function graph lies above the power function graph, except between two values of *x*? Explore by dragging the red vertex of the slider *a* in the sketch below. Can you make a conjecture? (Note: you can also change the scale of the graph on each of the axes by dragging points *A* and *B*.)

####

The graphs of *a*^{x} and *x*^{a}

Check your conjecture by clicking here.

This page uses **JavaSketchpad**,
a World-Wide-Web component of *The Geometer's Sketchpad.*
Copyright © 1990-2011 by KCP Technologies, Inc. Licensed only for
non-commercial use.

*Back
to "Dynamic Geometry Sketches"*

*Back
to "Student Explorations"*

By Michael de Villiers. Created, 10
May 2014.