You're good to delete log files. They mostly contain information about failed actions (Failed to open file X, failed to save file Y, couldn't open module Z) along with other misc information. As a rule, anything that is *.txt, *.log won't seriously affect program behaviour. Those files are just plain old regular text.
Avast! is a little misleading with the advertising though. You will see practically zero benefit from deleting those files unless you have a very large quantity of those log files (>10-25GB). Even then, you're likely not going to see a huge benefit.