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Logarithms: History and Use

Date: 7/12/96 at 16:50:38
From: Linda Temple
Subject: Logarithms: Why they Work, History, and Name

I have been asked to explain logarithms from a non-numerical sense to 
non-math-oriented people.  It doesn't seem to be enough for me to show 
the equation and how it works, they want to know why. Any thoughts?

Also, do you have short anecdotal history for the development of the 
concept of logarithm?

Finally, why is it called a "logarithm"?  logos = reason, arithmos = 

Date: 7/13/96 at 12:21:19
From: Doctor Anthony
Subject: Re: Logarithms: Why they Work, History, and Name

It is a very great economy of effort if we can reduce multiplication 
to the addition of two numbers. The possibility of adding numbers 
that can be looked up in tables compiled "forever," as Napier 
remarked, instead of carrying out a lengthy process of multiplication, 
was suggested in two ways that were quite independent.  The first 
arose in connection with the preparation of trig. tables for use in 
navigation. The second was closely connected with the laborious 
calculation involved in reckoning compound interest on investments.

In 1593 two Danish mathematicians suggested the use of trig. tables 
for shortening calculations. They used the formula:

sin(A)*cos(B) = (1/2)sin(A+B) + (1/2)sin(A-B)

Thus to multiply 0.17365*0.99027, you look up in tables and find
0.17365 = sin(10),  0.99027 = cos(8)

and the above formula gives sin(10)*cos(8) = (1/2)(sin(18) + sin(2))

From tables sin(18) = 0.30902      sin(2) = 0.03490

sin(18) + sin(2) = 0.34392  and (1/2)(sin(18)+sin(2)) = 0.17196

Giving  0.17365*0.99027 = 0.17196

This device probably suggested to Napier, who is usually called the 
inventor of logarithms, a simple method for multiplying by a process 
of addition. 

Napier had been working on his invention of logarithms for twenty 
years before he published his results, and this would place the origin 
of his ideas at about 1594. He had been thinking of the sequences 
which had been published now and then of successive powers of a given 
number. In such sequences it was obvious that sums and differences of 
indices of the powers corresponded to products and quotients of the 
powers themselves; but a sequence of integral powers of a base, such 
as 2, could not be used for computations because the large gaps 
between successive terms made interpolation too inaccurate. So to keep 
the terms of a geometric progression of INTEGRAL powers of a given 
number close together it was necessary to take as the given number 
something quite close to 1.  

Napier therefore chose to use 1 - 10^(-7) or 0.9999999 as his given 
number. To achieve a balance and to avoid decimals, Napier multiplied 
each power by 10^7. That is, if N = 10^7[1 - 1/10^7]^L, then L is 
Napier's logarithm of the number N. Thus his logarithm of 10^7 is 0.  
At first he called his power indices "artificial numbers", but later 
he made up the compound of the two Greek words Logos (ratio) and 
arithmos (number).

Napier did not think of a base for his system, but nevertheless his 
tables were compiled through repeated multiplications, equivalent to 
powers of 0.9999999   Obviously the number decreases as the index or 
logarithm increases. This is to be expected because he was essentially 
using a base which is less than 1. A more striking difference between 
his logarithms and ours lies in the fact that his logarithm of a 
product or quotient was not equal to the sum or difference of the 
logarithms.  If L1 = log(N1) and L2 = log(N2), then 
N1 = 10^7(1-1/10^7)^L1  and N2 = 10^7(1-1/10^7)^L2, so that 
N1*N2/10^7 = 10^7(1-1/10^7)^(L1+L2), so that the sum of Napier's 
logarithms will be the logarithm not of N1*N2 but of N1*N2/10^7.  
Similar modifications hold, of course, for logarithms of quotients, 
powers and roots. These differences are not too significant, for they 
merely involve shifting a decimal point.

Napier's work was published in 1614 and was taken up enthusiastically 
by Henry Briggs, a professor of Geometry at Oxford. He visited Napier 
and discussed improvements and modifications to Napier's method of 
logarithms. Briggs proposed that powers of 10 should be used with 
log(1) = 0 and log(10) = 1. Napier was nearing the end of his life, 
and the task of making up the first table of common logarithms fell to 
Briggs. Instead of taking powers of a number close to 1, as had 
Napier, Briggs began with log(10) = 1 and then found other logarithms 
by taking successive roots. By finding sqrt(10) = 3.162277 for 
example, Briggs had log(3.162277) = 0.500000, and from 10^(3/4) = 
sqrt(31.62277) = 5.623413 he had log(5.623413) = 0.7500000.  
Continuing in this manner, he computed other common logarithms.  
Briggs published his tables of logarithms of numbers from 1 to 1000, 
each carried out to 14 places of decimals, in 1617. Briggs also 
introduced the words "mantissa" for the positive fractional part and 
"characteristic" for the integral part (positive or negative).

The first tables of logarithms contained inaccuracies which were 
noticed and corrected from time to time. The labor expended in 
constructing them was enormous, and it stimulated the search for 
better methods of calculating them. This gave a new impetus to the 
study of infinite series, for example sqrt(2) = (1 - (1/2))^(-1/2) 
which gives rise to an infinite, convergent series when expanded 
according to the binomial theorem. This work culminated in the 
extremely important exponential series:
where e = Limit {1 + 1/n}^n  as n -> infinity.  It is easy to show 

e^x = Limit {1 + 1/n}^(nx)     generates the series shown below:

e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! + ... to infinity, and

e = 1 + 1 + 1/2! + 1/3! + 1/4! + ....  = 2.718281828...

e is now used as the base of logarithms in almost all advanced work.

-Doctor Anthony,  The Math Forum
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Associated Topics:
High School History/Biography
High School Logs

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