As of patch 5.18, which was in mid-September, punishments are given out in this order:
1.) 10 game chat restriction
2.) 25 game chat restriction
3.) 14 day ban
4.) Permanent ban
The way it works is once you receive one punishment, your next offense will give you the next highest tier of punishment, regardless of how "minor" your offense may have been. It is possible for Riot to escalate you straight to a 14 day or permanent ban, but Riot really only does this if they deem your case severe enough. Also, it is possible to lower your punishment level by one tier if you go 3 months without receiving any punishments.
Something to keep in mind, too, is that even though you were only sent chat logs for 1-3 games, those chat logs are not necessarily the only ones you were punished for. The system doesn't send you all of the chat logs that you were punished for, only the most recent. It is possible to be punished for only one game, but it has to be very extreme for that to happen.
So, my question to you is, have you received a 25 game chat restriction within the past 3 months? If you have, then the 14 day ban is standard for the next punishment. If you haven't, then it's quite likely that Riot deemed your case severe enough to escalate you straight to a 14 day ban.
Now, if you believe that you should not have received a 14 day ban, your best bet is to submit a support ticket here to talk with Riot directly about your punishment. It's quite rare, but it is possible there was a mistake made on Riot's end, and I'm positive that Riot is more than willing to fix it if there was. Note that it may take a while to get a response, so you will need to be patient.