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### Natural Logarithms

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Date: 11/01/97 at 15:11:07
From: MWaissblut
Subject: Natural Logarithms

What is "natural" about natural logarithms? Also, why is 'e' a
transcendental number?
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```
Date: 11/02/97 at 07:28:20
From: Doctor Jerry
Subject: Re: Natural Logarithms

Hi,

I think that there is no single answer as to why natural logarithms
are preferred. Here are several possible answers.

1. Just as radians are preferred in differentiation, to avoid
constants, the exponential function e^x is preferable to, say,
10^x.  The derivative of e^x is, of course, e^x while the
derivative of 10^x is (10^x)*ln(10).

2. The differential equation dQ/dt=k*Q giving the rate of decay of a
radioactive material separates as dQ/Q = k*dt and then the solution
has the form ln(Q)=k*t+c. This is related to the fact that the area
beneath the curve y=1/x, from x=1 to x=t, is ln(t).

3. Related to number 2, if one writes the expression for the principal
at time t of an amount invested at r percent, compounded k times
per year, and takes the limit as k becomes infinite (to obtain
continuous compounding), it happens that one obtains an expression
of the form (1+1/m)^m. The question is what is the value of this
expression as m becomes infinite. The answer is e.

The point is that the number e occurs "naturally" in calculations
dealing with growth, just as pi occurs naturally in connection with
circles or spheres.

As to why e is transcendental? Why is pi transcendental? To give an
answer runs the risk of going philosophical, which, in my opinion,
rarely leads to useful thoughts. However, there would seem to be no
reason that pi should turn out to be 22/7, for example. Pi can be
thought of as the result of any of several infinite processes
(inscribing polygons in a unit circle, for example) and I think one
would be surprised if a rational number resulted. Since the
exponential function and trig functions are so strongly related, it
would be surprising if e were to be rational.

-Doctor Jerry,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
High School Logs
High School Number Theory

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