Logarithms and Base E

Date: 11/20/97 at 00:31:33
From: Steven
Subject: Logarithms and base e

I cannot figure out why the base of a natural logarithm is "e" - I 
have asked my professor, but he was leaving for the day and said that 
"it is a long and complicated process to prove why it works." My 
question was how did "e" recive a value of 2.17... ? Can you help me?

Date: 11/24/97 at 16:05:43
From: Doctor Bruce
Subject: Re: Logarithms and base e

Hello Steven,

No doubt you are familiar with common logarithms, which are to the 
base 10. These seem natural enough to most poeple, since we are 
familiar with arithmetic in base 10. It sounds as if you want to know 
where the very strange-looking number  2.7182818284 ...  comes from, 
and why we mathematicians think it is so much more natural than good 
old 10. 

If we plot the graph of the common logarithm function  y = log_10(x) 
and take some very careful measurements, we find that the slope of the
tangent line at the point  (1,0)  is approximately equal to .434.

We can plot the logarithm function with other bases, too. For example, 
if we plot y = log_2(x), we find the slope of the tangent at (1,0) is 
approximately equal to 1.44. If we plot y = log_3(x), we find the 
slope of the tangent at (1,0) is approximately equal to .91.

So, we ask, is there a value between 2 and 3 which could serve as the 
base for logarithms, such that the tangent at (1,0) has slope exactly 
equal to 1? The answer is yes, that value is  2.718 ..., which we 
typically call "e".

If you know some calculus, in particular, how to take derivatives, 
then you know that the derivative of  log_e(x) is 1/x. But for any 
other base of logarithms, the derivative will be 1/x multiplied by a 
number other than 1. The derivative of log_2(x) is (1.44...)/x, for 
example. That's why using  e  is so natural.

The value of e is easily computed to as many decimals as you like by
adding up a few terms of the series

     e  =  1 + 1/1! + 1/2! + 1/3! + 1/4! + ...  

-Doctor Bruce,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 11/25/97 at 13:42:24
From: Anonymous
Subject: Re: Logarithms and base e

Hey thanks, now I know where e got its value.