will the F5 log incoming connections into a log file?

Hi,

BIG-IP system is not a logging server and has limited capacity for storing, archiving, and analyzing logs. A dedicated logging server is recommended for extensive logging functions. The BIG-IP system is configured by default to provide the most relevant log information to administrators.

Changing the default log levels to a higher level increases the amount of data stored on the device. If the default levels are changed for troubleshooting purposes, remember to set the level back to its default setting.

Additional to modify the log setting for default you to have to go to the path:

System ›› Logs : Configuration : Options

Additional you can export the log connection to a file the  inside your virtual server with this command:

tmsh show sys connection cs-server-addr 192.168.138.30 > /var/tmp/connection.txt

Hope it´s work.