Natural Logarithms

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? 

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/