The Logarithm log(x)

Abstract

We return to the question of the existence of a primitive function of f (x) = 1/x for x > 0 posed above. Since the function f(x) = 1/x is Lipschitz continuous on any given interval [a, b] with 0 < a < b, we know by the Fundamental Theorem that there is a unique function u(x) which satisfies u′(x) = 1/x for a ≤ x ≤ b and takes on a specific value at some point in [a, b], for example u(1) = 0. Since a > 0 may be chosen as small as we please and b as large as we please, we may consider the function u(x) to be defined for x > 0.

Nevertheless technicalities and detours should be avoided, and the presentation of mathematics should be just as free from emphasis on routine as from forbidding dogmatism, which refuses to disclose motive or goal and which is an unfair obstacle to honest effort. (R. Courant)