## An interesting property of the number e

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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.

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By Michael de Villiers. Created, 10
May 2014.