This applet compares the values of [math]k^x[/math] and [math]x^k[/math]. When k=e, the graph is non-negative, and k=e is the only value for which [math]k^x≤x^k[/math] for all x.
A nice activity for exploring exponents is to have students play a finger game where (like "Rock Paper Scissors") they each hold out 1 to 5 fingers. Their own number is the base, the other player's the exponent, highest score wins (e.g. if I hold 5 fingers and you 2, then you win because 5^2<2^5). Then extend and allow each person to write any positive number on a piece of paper.
This applet is easy enough to make yourself -- I put it up as part of an article I'm writing about several related problems. I am new to GeoGebra Tube, and if there are two copies, it's because I'm trying to get it the right shape for embedding on my site.
