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The Number e and the Natural Logarithm - Concept
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
The natural log is the logarithm to the base of the number e and is the inverse function of an exponential function. Natural logarithms are special types of logarithms and are used in solving time and growth problems. Logarithmic functions and exponential functions are the foundations of logarithms and natural logs.
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.
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
When B equals 2. Again, the slope
When B is 3, the slope is
So that makes me wonder where
is the slope equal to one?
Is it 2.5? No.
It turns out that if I want to get the
slope to be exactly 1, I need B to be
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
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.
How about a million?
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.
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