A standard bridge between automata theory and logic is provided by the notion ofcharacteristic formula. This paper investigates this problem for the class ofevent-recording automata (ERA), a subclass of timed automata in which clocks areassociated with actions and that enjoys very good closure properties. We first study theproblem of expressing characteristic formulae for ERA in Event-Recording Logic (ERL ), alogic introduced by Sorea to express event-based timed specifications. We prove that theconstruction proposed by Sorea for ERA without invariants is incorrect. More generally, weprove that timed bisimilarity cannot in general be expressed in ERL for the class of ERA ,and study under which conditions on ERA it can be. Then, we introduce the logicWTμ , a new logic for event-based timed specificationscloser to the timed logic ℒν that was introduced byLaroussinie, Larsen and Weise. We prove that it is strictly more expressive than ERL , andthat its model-checking problem against ERA is EXPTIME -complete. Finally, we providecharacteristic formulae constructions in WTμ forcharacterizing the general class of ERA up to timed (bi)similarity and study thecomplexity issues.