Logarithms: History and UseDate: 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 = number. 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 that 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 Check out our web site! http://mathforum.org/dr.math/ |
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