My attempt there at a definition has not been satisfactory enough to me. Here I pursue another approach.
We begin with $$\mathcal{P}=2^S,$$ which is symbolic but we try to assume more in a correspondence principle:
to find insight for new definitions in set theory, we seek help from relations between cardinals of sets.
Since $$|\mathcal{P}|=2^{|S|},$$
$$|S|=\log_2 |\mathcal{P}|$$
the proposed correspondence principle leads us to consider the question
given the power set of a set, how do we find the set itself?
The answer is simple: find the singletons of \(\mathcal{P}\). That is
$$S =\log_2 \mathcal{P}= \{x \mid \{x\} \in \mathcal{P}(S)\},$$
therefore
$$\boxed{\log_2 A := \{x \mid \{x\} \in A\}}$$
