LOGARITHMS. A tabular system of num bers, by which multiplication can be performed by addition, division by subtraction, involution by a single multiplication, and evolution by a single division. The logarithm of a number is the exponent of the base which produces the number. if am = b, x is said to be the logarithm of b to the base a. Any finite positive number greater than 1 may he taken for a base, and the logarithms of all positive numbers with respect to the base may be tabulated. The base 10. how ever, has been found the most convenient. and the system of 'common logarithms' constructed upon it is universally used for arithmetical computa tions. The following brief table will serve for purposes of illustration: Here 3 means —3, and 3.3010 means —3 0.3010. According to the definition, 3 is the logarithm of 0.001, written log 0.001 = log 0.002 = 3.3010 . . . log 10,000 = 4. The integral part. of the logarithm is called the characteristic and the decimal part the mantissa. The charac teristics of the logarithms of positive numbers less than 1 are negative, of numbers not less than 10 positive, and of numbers between 1 and 10 they are zero. The mantissas are always taken as positive and are generally incommensurable. A table of mantissas expressed to six decimal places is sufficiently accurate for all ordinary purposes. In the common system, it is unneces sary to tabulate the characteristic, since its value is always one less than the number of places to the left of the decimal point. For example, log 059.34 = 2.819109. Another advantage of the common system is that the mantissas of the loga rithms of numbers having the same sequence of figures are equal. For example, log 659.34 = 2.819109 and log 65934 = 4.819109. These two properties belong to the common system only, and to them it owes is superiority over others for the purposes of numerical calculations.
The logarithm being an exponent, it must obey the laws of exponents, and from these are de rived the fundamental principles of logarithmic calculation: (1) The logarithm of a product is equal to the sum of the logarithms of its fac tors. (2) The logarithm of the quotient of two numbers is equal to the logarithm of the dividend less the logarithm of the divisor. (3) The loga rithm of a number affected by an exponent is equal to the exponent times the logarithm of the number. E.g. to multiply 50 by 70, log (70.50) = log 70 log 5 + log 10 = I.8451 + 0.6990 + 1= 3.5141 from the above table. But the num ber corresponding to the log 3.5441, called the antilogarithm is 3500. 70 .50 = 3500. Also, to extract the square root of 4900, log V4900 or log 49001 = V. log 4900 = . 3.6902 = 13451 from the table. But the antilog 1.8151 is 70, 4900 = 70. In finding the logarithm of a quotient, especially when the divisor con tains several factors, it is easier to add the com plement logarithm or cologarithm of the divisor than to subtract its logarithm. The cologarithm
of a number is defined as the logarithm of its reciprocal. The colog u = log — 1 = — log n,n which justifies its use as explained.
John Napier (1614), a Seotelnan. is usually regarded as the inventor of logarithms, although Biirgi (1552-1632) had probably as early as 1607 computed a table of antilogarithms. but he did not fully understand the importance of the in vention, and failed to make it public until 1620, when Napier's logarithms were known throughout Europe. Bfircd's drithinetisrhc nod Geometrisehe Progress-TalLen was published in 1620 in Prague, and contains logarithms of ordinary num bers. while Napier's Mirifiri Logarithmorum Pa rotas Dekriptio contains logarithms of trigono metric functions. ( Sec TRIGONOMETRY.) Tables of the numerical values of the trigonometric func tions had attained a high degree of accuracy at this time, but their usefulness depended upon abridged methods of calenlation. and the search after such methods led to the discovery of loga rithms. Napier observed that if in a circle with the radii OA, (r = 1) at right angles, the sine parallel to OA, moves from 0 to at intervals forming an arithmetic progression, its value decreases in geometric progression. The segment Napier originally called numerus artilicialis, and later logarithm's (ratio num ber). The first computers of logarithms did not understand the connection between logarithms and exponents. but modern investigations show that 2.7184593 is the base of Biirgi's logarithms. and that is the base of (1619. 1624), in adapting Naperian loga rithms to positive integers, employed as a base 2.718281828. known since time time of Euler as e. This system was called by Halley the Nape rian system. and this name has been retained, so that to-day `Naperian logarithms' mean loga rithms to the base c. Such logarithms are called anlural logarithms, and the relation between Naperian logarithms of a sine S and its natural logarithm is expressed by the equation Nap. log 10' S = 10' nat. being taken as the sine of 90' and its logarithm zero. According to the integral calculus, the equation f = log x + lc may serve as a definition for the logarithm of a number, and from it is derived a relation which gives to the natural logarithm the name hyper bolic logarithm. The equation of a rectangular hyperbola (q.v.) referred to its asymptotes is .ry = c, hence y = - , and yclx= edx a n element of area of base dx and altitude y between the curve and the x-axis; the area between the curve, the x-axis, and two ordinates at is propor tional to log It appears from the definition that logarithms formed from one base must bear a constant ratio to those formed from another base. This ratio is called the modulus of the first system. The modulus of the common system is 0.43429448 . . . hence if the hyperbolic logaritlun of a number is I. its common logarithm is 0.434294481.