Changing the log format is in what -- SDK? XML? (Sorry -- I've never dealt with this, neither do I know how to work with SDK).
Depends on what you want to change. If you just want to change the wording, you can do that in the XML. I don't know the tag name for it, so you'll just have to hunt around the game text files. If you want to change the information it provides you with, that will probably be in the SDK.
So, for example... in the XML, you'll see something like this:
Our %s1_Unit has been killed by a %s2_Adj %s3_Unit.
In the SDK, you basically provide it with the information you want... in this case it would be the namekey for our unit, the name key for the opposing player, and the name key for their unit.
If you want it changed to something like this:
Our %s1_Unit was destroyed by an %s2_Adj %s2_Unit.
You could do that in the XML. You can also choose to omit information too in the XML without needing SDK changes. If you delete the killing player's adjective, it won't cause problems.
But let's say you want it to provide the name of the city that's nearest to the tile:
Our %s1_Unit has been killed by a %s2_Adj %s3_Unit near %s4_City.
That would require SDK work because you'd have to find the nearest city and then tell the program to include that information in this message.