BIG-IQ collecting BIG-IP iRule logs

BIG-IQ DCD's (Data Collection Devices) can act as repositories for 'alerts, events, and statistical data from one or more BIG-IP systems', but do not store general system logs. BIG-IQ can help you create logging profiles which can be deployed across one or more BIG-IP systems and can have filters which match log messages of interest, but you need to have some remote storage and viewing system.

Here's the documentation for creating and deploying logging profiles:

https://techdocs.f5.com/en-us/bigiq-7-1-0/managing-ltm-and-network-objects-using-big-iq/managing-logs.html