Natural Logarithms

Date: 11/23/98 at 16:43:10
From: Chris Harvey
Subject: Origin of the name "natural logarithms"

I have perused many math history Web sites but have been unable to 
find the origin of the name "natural logarithms." I am aware that 
logarithm is derived from two Greek words, but what is the significance 
of "natural"?  Why did logarithms to the base e receive the name 
"natural"?

Date: 11/25/98 at 12:44:52
From: Doctor Jaime
Subject: Re: Origin of the name "natural logarithms"

Hello Chris,

The base of the logarithms that were considered most important has 
varied throughout history. Napier or Neper, the inventor of logarithms, 
did not even bother with this and, although some mistakenly say  e is 
the base of Neperian logarithms, this is incorrect because the 
logarithms used by Napier, Nap.log, are related to natural logarithms 
in this strange way:

                          10^7
   Nap.log x = 10^7 * ln -------          
                            x

The first mathematician to use the name "natural" was N. Mercator. He 
also used the word "hyperbolic" for the first time for logarithms, and 
in 1668 he showed that:
                              
    /x
    |    dt
    |  ------  
   /1    t
                             
only differs from the logarithm in base 10 by a constant factor. The 
base e  became the most important one (if you search our archives you 
will find lots of reasons for that) and the name "natural" became more 
popular than the name "hyperbolic" although in some countries the name 
"neperian logarithm" is also used.

You can find more about Napier and N. Mercator in the St. Andrews 
MacTutor History of Mathematics archive:

  http://www-history.mcs.st-and.ac.uk/history/Indexes/Full_Alph.html   

Feel free to write us back if you have more questions.

- Doctor Jaime, The Math Forum
  http://mathforum.org/dr.math/