  The Shannon entropy of a random variable $X$ has much behaviour analogous to
a signed measure. Previous work has concretized this connection by defining a
signed measure $\mu$ on an abstract information space $\tilde{X}$, which is
taken to represent the information that $X$ contains. This construction is
sufficient to derive many measure-theoretical counterparts to information
quantities such as the mutual information $I(X; Y) = \mu(\tilde{X} \cap
\tilde{Y})$, the joint entropy $H(X,Y) = \mu(\tilde{X} \cup \tilde{Y})$, and
the conditional entropy $H(X|Y) = \mu(\tilde{X}\, \setminus \, \tilde{Y})$. We
demonstrate that there exists a much finer decomposition with intuitive
properties which we call the logarithmic decomposition (LD). We show that this
signed measure space has the useful property that its logarithmic atoms are
easily characterised with negative or positive entropy, while also being
coherent with Yeung's $I$-measure. We present the usability of our approach by
re-examining the G\'acs-K\"orner common information from this new geometric
perspective and characterising it in terms of our logarithmic atoms. We then
highlight that our geometric refinement can account for an entire class of
information quantities, which we call logarithmically decomposable quantities.
