Example 1: The Natural Base | Click here to open an associated Mathcad worksheet: |
Plotted below are two familiar exponential functions:
On the interval shown, we see that 2^{x} begins with a slight head start in value, but 3^{x} quickly outgrows it, overtaking 2^{x} at
It is interesting to compare the average growth rates of these two functions at different points. We do so in the table below. In the table, values of f(x) and g(x) have been computed at equal intervals of
It appears that the average rates of change for f are always lagging a little behind the values of the function itself, while those for g are always a little ahead. We can see this clearly if we plot each of the functions together with their computed average rates.
Notice that, while neither curve's values coincide exactly with its growth rates, the match is a bit closer for
A reasonable question then seems to be: Is there a number, somewhere between two and three (and perhaps a bit closer to three), that is the base of an exponential function for which the growth rates are exactly the same as the values at any point? You've probably guessed the answer: The base is e ! Below, we plot
The exponential function y = e^{x} grows at a rate equal to its value. |
This is the basic example, and the reason for e's preeminent place in the study of exponential and logarithmic functions.
The next example shows how e may lay hidden, like a lion in tall grass, in many numerical series.