How can you retrieve a log of your iMessage s

If you use iMessage, those messages appeared in blue text bubbles are always encrypted. So you can not see any log from iMessage. If you want to find or recover your old conversation history, besides restore the iCloud backup files, you can try in the following way. Actually, the saved messages in iMessage are stored in a chat.db data base (~/Library/Messages), which allows you to see the "history" to any iMessages conversation on Mac pro. The chat.db holds all the information from your active message history with the clear labeled symbol. If this doesn't work for you, I afraid you might need to try some tool for help. Detailed introduction can be found here: How to recover deleted iMessages from iPhone.

iMessages (blue bubbles) do to go through your mobile providers system they go through Apple's servers. Your provider never sees them.

No, the only "log" is the iMessages themselves. As I said if you are doing regular backups you can retrieve deleted ones.

There is no such log. Did you accidentally delete your iMessages? If you have a backup you can restore from that and recover them.