Math Is Fun Forum

Chewy

Member

Registered: 2008-08-07

Posts: 67

Natural logarithm, "e"

Hi all,

I have searched the forum and have been having a hard time finding something in regard to what "e" represents. I followed a link from here to the Mathisfun.com site (http://www.mathsisfun.com/numbers/e-eulers-number.html), and I feel that I understand how to obtain e as a number. A very good explanation. One shortcoming on my part, is I still do not understand what exactly its purpose is.

I have read that it is a constant, and at P(0,1) on an XY graph, the slope is equal to y at that point exactly. All functions of x that are negative result in an extremely low Y coordinate, until reaching positive values of x, where a change in Y dramatically increases positive. I am just not understanding what (1,e) represents on that same graph.

I am studying calculus materials in my spare time, and have read of this important number, and want to understand it better. Any help would be very much appreciated!

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Natural logarithm, "e"

Here's a musical ditty on e.
http://members.aol.com/lizanya/sugar.html
The background is a pretty design like fractals.
e is also called Napier's constant.
hyperbolic cosine and sin are related?
Regular sine and cosine waves have been
demonstrated using i, e, and pi; this is
very weird stuff but was shown to me
in electrical engineering class.
I heard e shows up in snail's spiral, maybe?

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igloo myrtilles fourmis

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Natural logarithm, "e"

e has quite a few uses. In calculus terms, the reason it's so important is that f(x) = e^x is the only function whose derivative is equal to itself.

Therefore, g(x) = e^(kx), where k is some constant, is the function whose derivative is proportional to itself and so this is useful in modelling patterns of growth.

Say you have a population of antelope and want to know how many there will be in a few years.
It's reasonable to assume that the amount of baby antelope that get born in a given time will be some proportion of the amount that are already there, which means that the formula for the population in terms of time will involve e^x in some way. It's actually more complicated, because death and infancy need to be considered, but that just means more stuff in addition to the e^x.

It works the other way too. A block of radioactive material will decay at a rate proportional to how much of it is radioactive. We're talking decay now instead of growth, so the formula will involve e^(-x), but it's the same principle.

The page that you mentioned gives another use for e, in working out compound interest for bank accounts.

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Why did the vector cross the road?
It wanted to be normal.

Chewy

Member

Registered: 2008-08-07

Posts: 67

Re: Natural logarithm, "e"

Thank you Mathsy, that helps !

To John E: Interesting link lol

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Natural logarithm, "e"

f(x) = e^x is the only function whose derivative is equal to itself.

There are infinitely many real-valued functions whose derivative is equal to itself.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Natural logarithm, "e"

True point.
In those models I was talking about earlier, the value of A would be decided by how many antelope there were to start with, or how much radioactive stuff there was.

No matter what the starting value is though, the general trend of f(x) = Ae^x will always be the same, and two different A's would be indistinguishable if drawn on unscaled graphs.
(Except in the boring case when A=0 and nothing happens ever)

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Why did the vector cross the road?
It wanted to be normal.

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: Natural logarithm, "e"

(Except in the boring case when A=0 and nothing happens ever)

No, you don't mean this - the zero exceptions give color to maths. A space becomes funny when we find some distinguishable elements of it. The zero case often is called trivial, but it's not trivial at all - it's foundamental.

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IPBLE:  Increasing Performance By Lowering Expectations.

Chewy

Member

Registered: 2008-08-07

Posts: 67

Re: Natural logarithm, "e"

Please correct me if I am wrong..

If a curve on a graph is "cut" by a secant line (I didn't know what word to use), we could call one end of the secant point a and the other a+h, as h representing the difference in distance on the X axis of the two points.

If the line that is tangential to point a is pivoted on point a, thus decreasing the distance of a+h, this distance a+h could be termed a+h', the first derivative of a + h. Is there anything correct about this?

What I am trying to accomplish here is understanding derivatives better, and that's when I came across "e". Am I on the right track at all?

Chewy

Member

Registered: 2008-08-07

Posts: 67

Re: Natural logarithm, "e"

....anyone?

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: Natural logarithm, "e"

I think you've undestood, just one thing - we don't call the point a, every point on the curve is uniquely determined by its x-coordinate (there aren't 2 points with the same coordinate). And the derivative is not a distance, it's the slope of the line which approaches this so-called point a.

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IPBLE:  Increasing Performance By Lowering Expectations.

Chewy

Member

Registered: 2008-08-07

Posts: 67

Re: Natural logarithm, "e"

Thank you Krassi I appreciate the help !