There is good news and bad news, but I fear that the bad will outweigh the good.
The good news is that all Eraser tasks generate a log, though the log of tasks initiated by drag and drop or through the context menu is deleted immediately. But the logs of all tasks run from the schedule, including tasks that are run on demand, are maintained. Log entries can be highlighted and copied and pasted to text files, though this is a cumbersome process.
The bad news is that Eraser (both v5 and v6) logs are really designed for error reporting, and do not specify the actions taken. This, I think, rules them out of court for your purpose.
I can't know the logic of your requirement, of course, but there is an argument that logging erasures via the erasing program is in any case potentially exceedingly misleading. There are circumstances (e.g with a flash drive or SSD) where Eraser will report that it has successfully completed a file erasure when it has in fact done nothing of the kind. IMO, the only valid way to be sure that erasing has taken place is to run a file recovery program on the target drive. Sadly, I don't know one that generates a log of its scans.
I'm sorry I could not be of more help.
David