Logarithms (Gr. noyos, ratio, and api0uos, number), numbers so related to the natural numbers that the multiplication and division of the latter may be performed by addition and subtraction, and the raising to powers and the extraction of roots by multiplication and division of the former. The labor of these operations by the ordinary processes of arithmetic, when the numbers are composed of many figures, is enormous. By the use of logarithms, for the invention of which the world is indebted to John Napier of Merchis-ton, Scotland, this labor is greatly diminished. - The general theory of logarithms is very simple. All numbers whatever may be regarded as the powers of some other number taken as a base. Thus, taking as a base the number 8, its successive integral powers give the series of numbers 8, 64, 512, 4,096, etc.; for 81=8, 82=64, 83=512, 84=4,096, etc. But it is not necessary to limit the series to the integral powers. The cube root of 8=3√8= 81/2=2; the square of the cube root of 8= 3√/82=8 2/3=4. The first power of 8 multiplied by the cube root =8 x 81=81 1/3=8 4/3=16; 8 x 8 2/3 =81 2/3=8 5/3=32, etc. Other fractional powers would give the numbers omitted in this series; so that a power of 8 could be found which would be equal to any number whatever.

By taking negative powers, fractions would come into the series. In a system of logarithms of which 8 is the base, the logarithms are the exponents of the powers to which 8 must be raised to produce the number. Thus, as above, 1/3=log. of 2, 2/3=log. 4, l=log. 8, 4/3=]og. 16, 5/3=log. 32, 2=log. 64, 7/3=log. 128, etc. It is obvious that the base of the system may be taken to be any positive number except unity. To demonstrate the general principles of logarithms, let a represent the base of the system, m any number, and x its logarithm; then the relation between the number m and its logarithm is expressed by the equation ax-=m. In this equation, x when considered in its relation to a is called the exponent or index of a; when considered in its relation to m, it is called the logarithm of m. That is, the logarithm of a number is the exponent of the power to which the base must be raised to produce the number. Let m and n be two numbers, x and y their logarithms, and a the base; then ax=m; ay=n. Multiply the first members of these equations together, and we have ax x ay=az+y=mn; that is, x + y=log. mn, or the logarithm of the product of two numbers equals the sum of the logarithms of the numbers themselves.

Dividing the first of the equations above by the second, we have az/ay =m/n,or az-y =m/n that is, x - y= log. m/n, or the logarithm of the quotient of one quantity divided by another is equal to the logarithm of the dividend, less the logarithm of the divisor. In the equation ax+y= mn, if we make m=n, then x=y, and we have a2 x z=m2; 2x is then the logarithm of m2, or the logarithm of the square of a number equals twice the logarithm of the number itself. By similar reasoning it is shown that the logarithm of the cube of a number equals 3 times the logarithm of the number, etc. If we take m2=p, then m= √p=p1/2; but log. m2=2 log. m= log. p. Substituting in the last equation √p for m, it becomes 2 log. √p=log. p, or log. √p=1/2 log. p; i. e., the logarithm of the square root of a number equals half the logarithm of the number itself. In the same way it may be shown that the logarithm of the cube root of a number equals 1/3 the logarithm of the number, and the logarithm of any root of a number equals the logarithm of the number divided by the exponent of the root. - The system of logarithms in common use is that proposed by Henry Briggs, professor of geometry at Oxford, soon after the publication of Napier's invention in 1614. Briggs used as the base of his system the number 10, and it was soon universally accepted, being so well adapted to the decimal notation.

The logarithm of any number in this system is the exponent of the power to which the number 10 must be raised to produce the number. Thus, since (10)°=1, (10)1 = 10, (10)2 = 100, (10)3=1,000, (10)4 = 10,000, etc, 0, 1, 2, 3, 4, etc, are the logarithms respectively of 1, 10, 100, 1,000, 10,000, etc. A number between 1 and 10 will have for its logarithm a fraction between 0 and 1. Thus the log. of 2=0.30103, for (10)0.30103=2. A number between 10 and 100 will have for logarithm a number between 1 and 2; thus the logarithm of 50=1.69897, for (10)1.69897=50. Numbers between 100 and 1,000 will have for logarithms numbers greater than 2 and less than 3, or 2 plus a fraction; thus the log. 250=2.39794, for (10)2.39794=250, etc.-In order to make logarithms available for purposes of calculation, the logarithms of all numbers between convenient limits are computed and arranged in tables, the natural numbers occupying the leading or argument column, the logarithms being placed opposite in adjoining columns.

Sometimes tables are arranged with the logarithms in the leading or argument column; these are called tables of anti-logarithms. For certain purposes logarithms constructed substantially according to the system originally proposed by Napier are used, and are known as Napierian, natural, or hyperbolic logarithms. In this system the base is the number 2.7182818+ . These logarithms are of great use in the higher mathematics, and in the investigation of many problems in physics. The Napierian logarithm of a number is equal to the common or Briggs logarithm multiplied by 2.3025851, or divided by 0.4342945. - The early computers of logarithms carried them to ten places of decimals; but it was soon found that seven places were sufficient for most of the uses of astronomy, navigation, surveying, etc. In fact, five-place logarithms are often sufficient, and, being much more convenient and portable, should be used except when very great accuracy is required. The theory of logarithms is now taught as a part of liberal education, and is explained in all the treatises on algebra used in our high schools and colleges. Tables of logarithms are always preceded by directions how to use them.

For this purpose no knowledge of the theory is required, an acquaintance with the rules of arithmetic being all that is necessary. - An excellent collection of five-place logarithms is that attached to "Bowditch's Navigator," and also published separately under the title of "Bowditch's Useful Tables." This contains, besides the tables of logarithms for numbers, log. sines, tangents, etc, also many auxiliary tables useful in navigation and surveying. A good collection of five-place tables by J. Houel (8vo, Paris, 1858) contains also Gauss logarithms, so called from the name of their inventor. They are numbers by means of which, when the logarithms of two numbers are known, the logarithm of their sum or difference can be found without knowing the numbers themselves. Thus, suppose we have the log. of a and the log. of b, but do not know what numbers correspond to these logarithms, and we wish to know the log. of (a+b). "With the ordinary tables we should first have to find in the table the number corresponding to log. a, then the number corresponding to log. b, then add the two numbers together and find from the table the log. (a + b). This process requires three references to the table and one addition.

By means of a table of Gauss logarithms the same result is reached by one reference to the table and two additions. Among tables of logarithms to seven places of decimals may be mentioned Babbage's, which are very accurate. Taylor's tables (large 4to, London) are very valuable, but difficult to obtain. Shortrede's tables (large 8vo, Edinburgh) contain nearly all the tables required in computing; they are especially designed for military and civil engineers. The tables of Callet (8vo, Paris) are very good; they contain the logarithms of all numbers from 1 to 108,000, with log. sines, tangents, etc, besides tables of Napierian logarithms to 20 places of decimals, and short tables of common logarithms to 20 and to 61 places. For log. sines, tangents, etc, Bagay's tables (4to, Paris) are very convenient; they contain log. sines and tangents for every second of the quadrant. Of the tables recently published the most valuable are those of L. Schron, Siebenstellige Logarithmen (12th ed., Brunswick, 1873); Bruhns, "A New Manual of Logarithms to Seven Places of Decimals" (Leipsic, 1870), very beautifully printed; and G. von Vega, " Logarithmic Tables of Numbers and Trigonometrical Functions, revised and corrected by Bremiker" (55th ed., Berlin, 1873), which is very accurate and convenient.

The logarithms are given to seven places of decimals.