Complementary results to Heuvers's characterization of logarithmic functions (Q2404715)

\textit{K. J. Heuvers} [Aequationes Math. 58, No. 3, 260--264 (1999; Zbl 0939.39016)] proved a characterization of the logarithmic function. Now, the author obtains similar characterizations for multiplicative, exponential and additive functions, respectively. For example, one of the results states that, the function \(g:(0,\infty)\to (0,\infty)\) satisfies the functional equation \(g(x+ y)/g(x) g(y)= g(1/x+ 1/q)\) (\(x,y> 0\)) iff \(g\) is a multiplicative function. The proof is based among others on the result by Heuvers [loc. cit.].