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 Check out our web site! http://mathforum.org/dr.math/
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