Exponential and Logarithmic Functions
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
I want to talk about the natural logarithm. The natural logarithm is the inverse of the exponential function of base e. Let’s take a look at the number e again, Remember that it’s defined as the limit as n approaches infinity of 1 plus 1 over n to the nth power. And it’s approximately 2.71828.
So the problem here is to graph y equals e to the x and its inverse. Let me do that now, just really quickly. If I were to come up with a table of values for x and y equals e to the x, when I’m graphing any exponential function I tend to try to use easy numbers for x like -1 ,0, 1 only e is not such a nice number. It’s not very easy to do arithmetic with it.
However e to the 0, I know it’s going to be 1 and e to the 1 is going to be e, so that’s about 2.71828. E to the -1, you may have to do this on your calculator but it’s about .36. So when I plot these points, I just want to get them approximately right.
(-1, .36) it’s a little more than a third so I'll put it about here. (0, 1) goes here and 1, e2.7 it’s not quite 2 and 3/4, so just a little shorter of 2 and 3/4. That’s where it would be. So I just draw a smooth curve connecting these points and I’ll have my y equals e to the x.
Now I want to graph the inverse. All I do is I take each of these points that I plotted and I interchange the x and y coordinates. So instead of plotting (-1, .36) I plot (.36, -1).
So (.36, -1) would be about here. Instead of plotting (0,1) I plot (1,0). (1,0) would be here. And instead of plotting (1, 2.7), I plot (2.7, 1). So 2.7 is about here, 1, so there I have three points for the inverse of y equals e to the x and I connect them with a smooth curve.
And as you remember about inverses, I can write an equation for this; x equals e to the y. But it’s pretty clear that this curve defines a function. So I want to be able to write this equation in terms of x, y in terms of x. So I need to come up with some new notation. Y equals ln x means x equals e to the y. This is the definition of the natural log. Y equals the natural log of x means x equals e to the y. This defines natural log as the inverse of the exponential function with base e.
Always remember that the natural log is the log base e of x but it's used so often that it's got its own special symbol ln.