Page:The American Cyclopædia (1879) Volume X.djvu/583

LOGARITHMS 577 Toledo, Wabash, and Western, the Detroit, Eel River, and Illinois, and the Logansport, Craw- fordsville, and Southwestern railroads meet here. The city is surrounded by a rich agri- cultural region, and has an important trade. Considerable quantities of poplar and black walnut lumber are shipped. Water power is abundant, and is used to some extent. The principal manufactories are the extensive car works of the Pittsburgh, Cincinnati, and St. Louis railroad, a hub and spoke factory, and large founderies. There are three banks, gra- ded public schools, including a high school, and two daily and five weekly newspapers. LOGARITHMS (Gr. Adyof, ratio, and apidpos, 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 arith- metic, 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 re- garded 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, &c. ; for 8 J =8, 8 2 =64, 8 3 =512, 8 4 =4,096, &c. But it is not necessary to limit the series to the integral powers. The cube root of 8= 8 ^8= 8^=2; the square of the cube root of 8= 8 |/8 2 =81=4. The first power of 8 multiplied by the cube root =8 x 8S=8i$=8J=16 ; 8 x 81 =8ii=8i=32, &c. 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 ex- ponents of the powers to which 8 must be raised to produce the number. Thus, as above, i=log. of 2, =log. 4, l=log. 8, =log. 16, f =log. 32, 2=log. 64, =log. 128, &c. 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 loga- rithms, let a represent the base of the system, m any number, and x its logarithm ; then the relation between the number m and its loga- rithm is expressed by the equation a x 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 w, 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 loga- rithms, and a the base; then a x =m ; av=n. Multiply the first members of these equations together, and we have a x xa,y=a x +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 a" m m second, we have = , or a x ~y = ; that is, 171 xy= log. , or the logarithm of the quotient of one quantity divided by another is equal to the logarithm of the dividend, less the loga- rithm of the divisor. In the equation a x +v= mn, if we make m=n, then x=y, and we have a 2x *=w 2 ; 2x is then the logarithm of m 2 , 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 loga- rithm of the number, &c. If we take mtp, then m=^p-=.p but log. m 2 =2 log. m-=. log. p. Substituting in the last equation fp for m, it becomes 2 log. 4/j=log. p, or log. i/p% 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 ^ 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 loga- rithms in common use is that proposed by Henry Briggs, professor of geometry at Ox- ford, 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)=1,000, (10) 4 = 10,000, &c., 0, 1, 2, 3, 4, &c., are the logarithms respectively of 1, 10, 100, 1,000, 10,000, &c. A number between 1 and 10 will have for its logarithm a fraction between and 1. Thus the log. of 2=0-30103, for (10)- 30103 =2. A number between 10 and 100 will have for loga- rithm 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 (10f 39794 =250, &c.- In order to make logarithms available for purposes of calculation, the logarithms of all numbers between convenient limits are com- puted and arranged in tables, the natural num- bers occupying the leading or argument col- umn, the logarithms being placed opposite in adjoining columns. Sometimes tables are ar- ranged with the logarithms in the leading or argument column; these are called tables of anti-logarithms. For certain purposes loga- rithms constructed substantially according to the system originally proposed by Napier are