I'm not sure what type of logs you are referring to, so I'll answer this in two ways.
For syslogs, you can simply have 2 syslog servers defined on each network device. One pointing to Cisco DNA, the other to LogRythm.
More generically, there are APIs you can take advantage of to pull all sorts of information from Cisco DNA. For example, some folks put together a way to collect some information and put it into Splunk: https://developer.cisco.com/network-automation/detail/98cc1342-5b6c-11eb-b2c1-0ec2761e2c74/
I don't see any preprepared scripts for LogRythm though.