I am being asked what a natural logarithm is?  I have search every
resource i have, but have not found a simple explination.  I know the
important stuff - inverse on e^x, base of e, etc.  My calculus
students want to know what it is and why it is?  HELP!! In simple
terms.

I will try to answer as simple as possible without over-simplication.

In many natural processes decay functions play an important role. Let
us consider the term "half-life" used in the phenomenon of
radioactivity.

"The measurement of half-lives of radioactivity in the range of
seconds to a few years commonly involves measuring the intensity of
radiation at successive times over a time range comparable to the
half-life. The logarithm of the decay rate is plotted against time,
and a straight line is fitted to the points. The time interval for
this straight-line decay curve to fall by a factor of 2 is read from
the graph as the half-life. If there is more than one activity present
in the sample, the decay curve will not be a straight line over its
entire length, but it should be resolvable graphically (or by more
sophisticated statistical analysis) into sums and differences of
straight-line exponential terms."

from:
( The Encyclopedai Britannica CD Rom Deluxe Edition 2001 )

The formulas to describe this are

(1) -dN/dt =lambda N 

(with lamda = decay constant, N = amount of particles and t=time

which can be solved like this:

(2) N(t)= N(0)e^-(lambda t)

Thatīs why the logarithms are called natural logarithms. They appear
in many natural processes.
There are many more examples where processes that can be described in
similar formula using exponential functions.
Examples are transportation of heat through a wall, growth of plants
or bacteria and many more.

I hope this helps to answer your question. If you need more
information please post a clarification request before rating my
answer.

till-ga


Search strategy:

Internal search function of the Encylopedia Britannica
and personal knowledge as a natural scientist

---

Request for Answer Clarification by jsc-ga on 16 Feb 2004 16:34 PST

I think that is a excellent example of how logarithms (actually,
exponential equations are used in life), but I am pretty sure that
natural logarithms got there name for other reasons than being used in
natural processes.  It still does not answer what a natural logarithm
(ln) is and why it exists.  Have you found anything else?  Thanks.

---

Clarification of Answer by till-ga on 16 Feb 2004 23:48 PST

All sources available state that there is indeed one main reason for
the name "natural".

As already mentioned by majortom-ga in his comment the source
( http://en.wikipedia.org/wiki/Natural_logarithm ) 
mentions that there can be another reason besides the one mentioned:

"What's so "natural" about them?
Initially, it seems that in a world using base 10 for nearly all
calculations, this base would be more "natural" than base e. The
reason we call ln(x) "natural" is twofold: first, the natural
logarithm can be defined quite easily using a simple integral or
Taylor series as will be explained below; this is not true of other
logarithms. Second, expressions in which the unknown variable appears
as the exponent of e occur much more often than exponents of 10
(because of the "natural" properties of the exponential function which
allow to describe growth and decay behaviors), and so the natural
logarithm is more useful in practice."
The Taylor series leading to the number "e" is given at:
( http://en.wikipedia.org/math/63c083c45a4b64f2f123047dda84f66a.png )

Another source suporting the main reason for the name:
"Natural logarithms and exponentials appear in a tremendous number of
mathematical models for real world phenomena; you will see them
frequently in engineering"
( http://www.ces.clemson.edu/ge/staff/park/Class/ENGR120/Handouts/Logarithms.html )

Sorry, but even with another extended search I canīt find any other
reason for the name "natural".

Besides I doubt if it will be possible to answer the question "and why
it exists" you posted in your clarification request". Itīs a more
philosophical question to find out WHY natural phenomena exist.

Again I hope that now you are satisfied with my answer.

till-ga

Search strategy

( ://www.google.de/search?q=%22natural+logarithm%22+name+reason&btnG=Google+Suche&hl=de&ie=UTF-8&oe=UTF-8
)

The Wikipedia has a fine entry on this, including a paragraph that
directly addresses the question, but I can't seem to phrase it clearly
enough myself, so I'm going to admit my own understanding is limited
and give you the benefit of my research:

Natural logarithm - Wikipedia
http://en.wikipedia.org/wiki/Natural_logarithm

I am sure this will be helpful.

Here is a precise mathematical definition.

Natural logarithms -
a) in the curve y = a^x, a = e when dy/dx = a^x. In other words, the
curve of y = e^x has a slope of e^x for all values of x.
The inverse of y = e^x is ln y = x. So, x = ln y is y = e^x relfected
in the line y = x.

b) the integral of 1/x. So, if dy = dx/x, y = ln |x| + C.

c) integral (no pun intended) to all natural systems, rates of growth,
population cycles, electricity, capacitance, magnetism.

d) y = ln x has the following properties:
? is defined for all values x > 0
? has a range of y is an element of all real numbers
? is concave down for x > 0
? is increasing for x > 0
? has dy/dx = 1/x
? has an x-intercept of 1
? has a y-intercept of negative infinity
? is converging for x in the interval (0, 1]
? can be written as y = log(e) x (i.e. logarithm with base e)

- supermacman-ga

This may or may not be touched on already, I didn't read the posts in
their entirety.  Here's as simple and concise a reason as possible:

Derivatives of exponential functions are proportional to the function
itself, or in symbols:

If y = b^x, then dy/dx = kb^x = ky , with k = lim [{b^h - 1}/h] as h
extends to infinity (this is based on applying the explicit definition
of the derivitive to the find the derivative of the function b^x)

This being the case, the question arose as to what value of B would
make the above equation especially convenient.  Choosing a B to make
the proportionality constant k equal 1 would the simplest (or most
"natural") choice.  So what value of b makes k = 1?  None other than
e.  So if the number e is chosen as the base, then the exponential
function is equal to its own derivative.

This is the only function that has this property.