We introduce a multi-agent logic of explicit, implicit belief and awareness with a semantics using belief bases. The novelty of our approach is that an agent's awareness is not a primitive but is directly computed from the agent's belief base. We prove soundness and completeness of the logic relative to the belief base semantics. Furthermore, we provide a polynomial embedding of the logic of propositional awareness into it, and a polynomial embedding of our logic into the logic of general awareness. Thanks to the latter embedding, we show that the satisfiability checking problem for our logic is PSPACE-complete. Finally, we extend it by the notions of public announcement and private belief expansion.