Logarithm Formulae

Date: 8/16/96 at 2:16:21
From: Anonymous
Subject: Logarithm Formulae

Dear Dr. Math:

I have been searching for a way to do logarithms longhand.  That is, 
given a number, I would like to be able to calculate its common (or 
natural) logarithm. A friend and I have researched many math books 
looking for the answer. 

The closest that we have come is a book that gives some historical 
background on logs. Even reading the section on Napier and Mr. Briggs 
does not enlighten us as to how we would go about actually computing a 
logarithm for ourselves (without looking them up in a list of already 
computed logs). My friend's interest is historical; he wants to 
emulate the antiquarians who worked by hand.  My desire is related to 
a class in the  C  programming language that I have just completed at 
Camden County College with Prof. R. W. Carney.  It would be a good 
exercise for me to be able to turn the formula into an algorithm and 
eventually to code it into a workable (and executable) program. We 
have been stymied in our attempts to derive the necessary formula from 
either High School texts or from a Frosh. Calculus text.

If someone can locate a working formula for computing common or 
natural (Napieran) logarithms, I would be truly appreciative.

Sincerely,
   Ed Johnson

Date: 8/16/96 at 17:42:52
From: Doctor James
Subject: Re: programming

There are two ways, neither particularly pretty, that come to mind.
The first is a physical method of making a logarithmic scale on some
medium, and manipulating that to return logarithms (i.e., Napier's
Bones, slide rules). But I don't think this is what you were 
interested in.

The other is a computational method. Given a first guess, you can use 
the Newtonian approximation method (or similar methods) to any desired 
accuracy. This is well suited for a computer algorithm.

Another which I just remembered is using the identity
log x = (ln x)/(ln 10)  (for base 10 logs). You can calculate the 
natural log of x (ln x) to any desired accuracy using the infinite 
series that represents ln x, which should be available in any 
comprehensive 1rst year calculus book: 

  ln(1+x) = x - x^2/2 + x^3/3 - x^4/4 + ...

-Doctor James,  The Math Forum
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