Hello,
How long is the Log storage period?
Is the log storage period unit capacity or time?
I think you can create a counter variable and use the command to clear the log after a certain period of time.
You can also store the log based on the available memory volume.
Additionally, you can use a script to send a message when a certain number of counts is reached, warning about the approaching limit of storage space.
See item 12.8. Interpreter mode log files in the Script Manual. it has a short comment about log.
There is no actual limit to the log, as far as I know. You will be able to find entries dating back to the production date of the robot in the log_history file saved within the robot system. 
You will however find that too many entries can “corrupt” the file somehow. I have seen this happen multiple times when a program is not made right and it skips a lot of waypoints due to overlapping blends for instance.
This creates an entry every time, and the log file will not be able to contain it.
It is not documented anywhere how many entries are permitted, though.