How to access Steam chat logs?
Solution 1:
This can be updated. Steam does log chats.
7 years later
https://help.steampowered.com/en/accountdata/GetFriendMessagesLog
Solution 2:
Steam does not log your chats, at least not on your local machine, though their new EULA agreement does allow them to review your chat logs, they are stored on their servers or hidden in encrypted files on your desktop as no one has yet discovered them.
It is possible to log your chat via 3rd party support however as it requires some of your steam details, I cannot guarantee it's security and therefore will not post a link to it.