The combat log, as is, provides for a very limited functionality. So limited, in fact, that it doesn't meet the minimum criteria to be a log when you need it to.

First and foremost: it resets when you leave a field. This includes jumping to another field and docking. This is a major issue.
When you bot explodes, and you would like to better know exactly what happened... You generally go to the log and see. That's kind of the point of a log, you go check it after the fact and you can know exactly what happened and when. The log being lost when your bot dies really defeats the point.

As for secondary concerns:
critical hit information is not shown on attacks where a shield is up and the attack is fully absorbed.
Damage absorbed/resisted would be very useful to have as well.

Some love to the UI for the log would help as well, but if It could stay up just like the assignment window, It would be enough.