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, ax and xa. 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.)

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The graphs of ax and xa

Check your conjecture by clicking here.

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By Michael de Villiers. Created, 10 May 2014.