The ambient logic has been proposed for expressing properties of process mobility in the calculus of Mobile Ambients (MA), and as a basis for query languages on semistructured data. To understand the extensionality and the intensionality of the logic, the equivalence on MA processes induced by the logic (=L) is compared with the standard MA behavioural equivalence and with structural congruence (an intensional equivalence, used as an auxiliary relation in the definition of satisfaction of the logic). The main contributions include a co-inductive characterisation of =L as a form of labelled bisimilarity, and axiomatisations of =L on the synchronous and asynchronous (finite) calculus. The study shows that, surprisingly, the logic allows us to observe the internal structure of the processes at a very fine-grained detail, much in the same way as structural congruence does. A spin-off of the study is a better understanding of behavioural equivalence in Ambient-like calculi. For instance, behavioural equivalence is shown to be insensitive to stuttering phenomena originated by processes that may repeatedly enter and exit an ambient.