LOGARITHMS. The common logarithm of a number is the index of the power to which 10 must be raised to be equal to the number. Thus !Os= 1,000, so that the logarithm of 1,000 (usually written log. 1,000) is 3. Now 10=-10, loo, 10.-- Low, and it is well known that etc., thus: Log. 0.001 = —3 Log. 10 = 1 Log. 0.01 = —2 Log. 100= 2 Log. 0.1 --= Log. 1,000 = 3 Log. 1 = 0 Log. 10,000 = 4 It is evident that the logarithm of any num ber greater than 1 and less than 10 is frac the logarithm of any number greater than 10 and less than 100 is greater than 1 and less than 2. Again,. the logarithm of any ,num her less than 1 is negative. The logarithms of numbers have been calculated by Napier, Briggs, Mercator, Newton, Leibnitz, Halley, Euler, L'Houillier, Vlacq, Sherwin, Gardner, Hutton, Taylor, Callet, Schron, Huntington, Moore and others. Of works giving tables of logarithms we may mention those to which the names of Hutton, Callet and Vega are respec tively attached. Chambers' Mathematical Tables is a useful little treatise; it gives loga rithms of numbers to seven places of decimals. Suppose we wish to know the logarithm of the number 18.1. In a book of tables we only find the fractional part of the logarithm, it is .257679. Now 18.1 is greater, than 10 and less than 100, so that its logarithm is greater than 1 and less than 2; hence log. 18.1=1257679. To give examples: Log. 18100 = 4.257679 Log. 1.81 = 0.257679 Log. 1810 = 3.257679 Log. 0.181 = T257679 Log. 181 =2.257679 Log. 0.0181 =7.257679 Log. 18.1 = 1.257679, Log. 0.00181 =7.47679 means —3 + 0.257679. (For a full ex planation of the finding of logarithms and nat ural numbers by tables consult treatises on trigonometry, etc.). The integral part of a loga rithm is called its characteristic, the fractional parts its mantissa. Logarithms make arith
metical computations more easy, for by means of a table of them the operations of multiplica tion, division, involution or the finding of powers, and evolution or the finding of roots, are changed to those of addition, subtraction, multiplication and division, respectively. For instance, if x and v are the logarithms of any two numbers, the lb? and 10v; now the product of these numbers is 10 x y, so that the loga rithm of the product of two numbers is the sum of the logarithms of the numbers. Again, the quotient of the numbers is 106•; so that the logarithm of the quotient of two numbers is the difference of the logarithms of the num bers. Again, raised to the nth power is so that the logarithms of the nth power of a number is n times the logarithm of the number. Again, the nth root of is so that the logarithm of the nth root of a num 1 ber is —th of the logarithm of the number.
Hitherto we have spoken of common loga rithms, which were invented by Briggs; their base, as it is called, is 10. Now logarithms were first used by Napier of Merchiston (see NAPIER, JOHN), and he employed a base which is smaller than 10, it is the number 2.7182818...., or the sum of the infinite series 2 + g.*.T-I-, etc. This base is denoted by E in mathematical treatises, and the Napier ian logarithm of any number, say 7, is log. r 7, to distinguish it from log. 7, which is the com mon logarithm, whose base is 10. The com mon logarithm of a number is found from the Napierian by multiplying by 0.43429448. Napierian logarithms are of great importance in mathematics.