Characterizing the Logarithm Through Continuity, Monotonicity, and Functional Equations

Summary: The logarithm function on the positive real numbers can be characterized in several ways. In this paper, we focus on classical characterizations based on the product rule or the power rule, and we improve upon some results by Fubini and Milkman. Our main result can be stated as follows: If a continuous or monotone function on the positive real numbers satisfies a power rule for both a negative and a positive exponent, then it is the logarithm function or the zero function. We also briefly consider the arithmetic logarithm function and its connection to the concept of arithmetic derivative.