IMO 1979 SL 14

Origin: GRE

Problem

Find all bases of logarithms in which a real positive number can be equal to its logarithm or prove that none exist.

Solution

Let us assume that a \inR \ {1} is such that there exist a and x such that x = loga x, or equivalently f(x) := ln x/x = ln a. Then a is a value of the function f(x) for x \inR+ \ {1}, and the converse also holds. First we observe that f(x) tends to −\inftyas x \to0 and f(x) tends to 0 as x \to1. Since f(x) > 0 for x > 1, the function f(x) takes its maximum at a point x for which f ′(x) = (1 −ln x)/x2 = 0. Hence max f(x) = f(e) = e1/e. It follows that the set of values of f(x) for x \inR+ is the interval (−\infty, e1/e), and consequently the desired set of bases a of logarithms is (0, 1) \cup(1, e1/e].