The two systems of logarithms for which extensive tables have been calculated are the Napierian, or hyperbolic, or natural system, of which the base is e, and the Briggian, or decimal, or common system, of which the base is 10; and we see that the logarithms in the latter system may be deduced from those in the former by multiplication by the constant multiplier 1/log(e)10, which is called the modulus of the common system of logarithms. The numerical value of this modulus is 0.43429 44819 03251 82765 11289 ..., and the value of its reciprocal, log^(e) 10 (by multiplication by which Briggian logarithms may be converted into Napierian logarithms) is 2.30258 50929 94045 68401 79914 .... The quantity denoted by e is the series, 1 1 1 1 1 + --- + --- + ----- + ------- + ... 1 1·2 1·2·3 1·2·3·4 the numerical value of which is, 2.71828 18284 59045 23536 02874 .... _The logarithmic Function._--The mathematical function log x or log(e) x is one of the small group of transcendental functions, consisting only of the circular functions (direct and inverse) sin x, cos x, &c., arc sin x or sin^{-1} x,&c., log x and e^(x) which are universally treated in analysis as known functions. The notation log x is generally employed in English and American works, but on the continent of Europe writers usually denote the function by lx or lg x. The logarithmic function is most naturally introduced into analysis by the equation _ / x dt | log x = ---, (x > 0). _/ 1 t This equation defines log x for positive values of x; if x <= 0 the formula ceases to have any meaning. Thus log x is the integral function of 1/x, and it can be shown that log x is a genuinely new transcendent, not expressible in finite terms by means of functions such as algebraical or circular functions. A connexion with the circular functions, however, appears later when the definition of log x is extended to complex values of x. A relation which is of historical interest connects the logarithmic function with the quadrature of the hyperbola, for, by considering the equation of the hyperbola in the form xy = const., it is evident that the area included between the arc of a hyperbola, its nearest asymptote, and two ordinates drawn parallel to the other asymptote from points on the first asymptote distant a and b from their point of intersection, is proportional to log b/a. The following fundamental properties of log x are readily deducible from the definition (i.) log xy = log x + log y. (ii.) Limit of (x^(h)-1)/h = log x, when h is indefinitely diminished. Either of these properties might be taken as itself the definition of log x. There is no series for log x proceeding either by ascending or descending powers of x, but there is an expansion for log (1 + x), viz. log (1 + x) = x - 1/2 x^2 + 1/3 x^3 - 1/4 x^4 + ...;