Forward the output of command tmsh show sys connection to External Server Logs

If your goal is to monitor all connection flows, like *every single connection* that's established, you can do this using iRules or LTM Policies. But you have to pick exactly what data is important to you. BIG-IP is a full 2-sided L7 proxy so you can log almost any piece of data that you want on either the serverside or clientside in Ethernet, VLAN, IP, TCP, SSL, HTTP, etc, as long as it's not encrypted. If you can give a specific list of data items you want to log, we can provide an example config to do that.

If you want to know periodically what is the count of connections, zamroni777's solution is awesome. BIG-IP also keeps track of this (connflow count) and other important things internally in the RRD graphs:

https://my.f5.com/manage/s/article/K50425247