EXPO'NENT and EX'PONEN'TIAL (from Lat. exponere, to set, forth, from ex, out + ponere, to put). An exponent, in the primitive sense, is a number-symbol which shows how many equal factors enter into a power; e.g. in 2', 3 is the exponent of 2; in a*, 5 is the exponent of a. The exponent affects only the letter or number adjacent to which it stands, ab' meaning abbb. While the various forms and the theory of expo nents have been matters of growth, the notation as now used was introduced into algebra by the mathematicians of the seventeenth century. Chu quet had (1434) used exponents, but not with the same significance as at present, and a step toward the theory of the subject, including the use of fractional exponents. had long before been made by Oresme (fourteenth century). Kepler (1619) speaks of Burgi as having written for x, x', e, e .... IR, lz, Is:,... whereas he himself prefers 1, it% however, wrote for and 30for 20.e. Ilarriot (q.v.) wrote x' for xx, x' for xxx. Wallis (1656) explained the expressions and xo as indicat 1 7 ing the same as . and The theory of ex ponents has gradually received extensions until it has become an important division of algebra.
The following equations show the meanings of various exponents: 1 a° = 1, al =a; a-2 cc° = a = ; a = f The fundamental laws of exponents in algebra are: ; a° = am-°; &eV am"; (ab)° ; for all values of In and a. These operationsare sub ject to the associative law (q.v.), =a(m+°)+P ; to the commutative law, = aP+°+., and to the distributive law, = (eV nr% In quaternions (q.v.) and certain other branches of modern mathematics the con ventions as to exponents differ.
Functions in which the variable or variables arc involved as exponents are called exponential functions; e.g. a' = c, = b, = 8. In the last example x evidently equals 3. When such equations cannot he solved by factoring, it is best to apply logarithms (q.v.). Thus in 2' = 80, log 80 x•log2 = 1ogS0, and x = = 6.322. The log 2 .1- X 3X 't series ex— 4- -I- —1 ' z ' 1.2 "“ is called the exponential series. If x be taken as 1, the series gives e= 2.718281823 . . . , the base of the hyperbolic logarithms (q.v.).