You can log locally for testing and debugging purposes, but you should use High Speed Logging to a syslog server if you are want to keep collecting logs.
The BigIP is not optimized for the significant I/O generated by high levels of local logging.
Use this to create a HSL configuration: https://github.com/brett-at-f5/tmsh/blob/master/create_hsl_log_publisher.tmsh
Use this to log the full proxy connection (Client --> VS : SNAT --> Pool Member) via HSL: https://github.com/brett-at-f5/irules/blob/master/logging/log_ip_nat_hsl_irule.tcl
You can modify the above for Syslog format.