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# The Number e and the Natural Logarithm - Problem 1

###### Norm Prokup

###### Norm Prokup

**Cornell University**

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.

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###### Norm Prokup

PhD. in Mathematics, University of Rhode Island

B.S. in Mechanical Engineering, Cornell University

He uses really creative examples for explaining tough concepts and illustrates them perfectly on the whiteboard. It's impossible to get lost during his lessons.

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