The Number e and the Natural Logarithm - Concept

Okay.

I wanted to talk about the number E and

why we use it as a base for exponential

functions, why is it so special.




So I've drawn a picture here.

The graph of an arbitrary exponential function

F of X equals B to the X, that's

this graph in purple.

And I've also drawn its tangent

line at the .01

Now what I'd like to do is I'd like

to explore the relationship between

the base B and the slope

of the tangent line.

In order to do that, I'm going to use a

demonstration from Geometer Sketch Pad.

Okay.




So you can see I've got graphed the function

G of X equals 2 to the X here.

It's actually B to the X. But I can

change the value of B to any value

I want. That graph is in red.

And the graph, the tangent

line, is in blue.

Right now the slope is .693

Let me move this tangent line around.




Notice as I move the tangent line it's

still tangent at the .01,

But as I move the tangent line

around, the base changes.

As I move it to the tangent line is

less steep, the base gets smaller.

As I move it to the tangent line is

more steep, the base gets bigger.

And if I moved it so that the tangent line

had a negative slope, the bases between

0 and 1. Okay.




Let's take a look at some

particular values.

When B equals 2. Again, the slope

is .693,

When B is 3, the slope is

1.099.

So that makes me wonder where

is the slope equal to one?

Is it 2.5? No.

2.75? No.

It turns out that if I want to get the

slope to be exactly 1, I need B to be

2.71828.

It's this number E. It's the only base

that will make it so that the tangent

line has a slope of exactly 1.01. Okay.




So let's summarize what we discovered.

If B is greater than 1, then the slope

of the tangent line is positive.

If B is between 0 and 1, then the

slope of the tangent is negative.

If you want the slope to be exactly

1, you need B to equal E. And E is

approximately 2.71828,

So that is a little glimpse into

what makes the number E special.




Now let me give you a definition for the

number E. E has a very complicated

definition. It's a limit as N approaches

infinity of 1 plus 1 over N to the N.

Now, to help you understand this definition

a little bit better, I'm going

to calculate some values for this expression

1 plus 1 over N to the N.




So I'll make a little table.

Let me start with the value 1. When

N equals 1, I get 1 plus 1 over 1,

2, to the 1. So I get 2. And

anything past that I'm going

to need my calculator.

So when I plug in 10, I'm getting 1 plus

1 over 10 to the 10th power, according

to my calculator it's approximately

2.5937. If I plug in 100, I get 1 plus 1 over

100 to the hundredeth power. It's approximately 2.7048.

I'm going to keep going up by powers of 10.

So a thousand, I get -- I'm not going

to write this out anymore.

2.71692.

How about a million?

2.71828.




So you finally get some convergence

once you get N out to a million.

It takes quite a while for this limit,

for this limit to start getting really

close, for this value to start getting

really close to E. But remember

that E is defined as the limit

of this expression.

So the value, these values are heading

towards E as N goes to infinity.