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 (xh − 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 x2 + 1⁄3 x3 − 1⁄4 x4 + ...;
the series, however, is convergent for real values of x only when x lies between +1 and −1. Other formulae which are deducible from this equation are given in the portion of this article relating to the calculation of logarithms.
The function log x as x increases from 0 towards ∞ steadily increases from −∞ towards +∞. It has the important property that it tends to infinity with x, but more slowly than any power of x, i.e. that x−m log x tends to zero as x tends to ∞ for every positive value of m however small.
The exponential function, exp x, may be defined as the inverse of the logarithm: thus x = exp y if y = log x. It is positive for all values of y and increases steadily from 0 toward ∞ as y increases from -∞ towards +∞. As y tends towards ∞, exp y tends towards ∞ more rapidly than any power of y.