I am given a set of properties for an unknown function $f(x)$. In particular, not constantly zero, not negative, additive and continues for any $x$ between 0 and 1.
I am asked to show equivalence relation between $f(x)$ and $\bar{f(x)}=-k \log_2(x)$ where $k$ is a constant. This seems the basic formula for the entropy for one random variable. I derived immediate conclusions from the properties (in particular the exponent of $x$ equals coefficient). However, it is still not clear how to prove this. How can I go from a vague definition for a function and prove its only unique form is logarithmic. In other words, what characterizes a logarithm function with coefficient?