Foreign Key constraints ensure the integrity of the data is maintained. If there was no FK in place. then it would be easy to get orphaned and inconsistent data. In your example, if you just update a username to be Bob from Terry then all the records associated with Terry in the log table will now have no link back to the users record ie the Terry records will be orphaned.
Soooooo, in answer to your question, you can drop it make your update but it will throw an error when you try and recreate the FK unless you update the log records too.
You can create these constraints with a cascading updates/deletes which will filter your changes down to the child tables but in this case, it looks like its turned off.
I'd suggest just doing the following
UPDATE Log set Username = Bob WHERE username = Terry
UPDATE user set username = Bob WHERE username = Terry.