Search integration logs

I would like to be able to search the integration logs so that if I find an error, I can search the case or record to quickly determine if it was ultimately successful. 

Currently, the logs are really hard to understand and use. Because we can't search, if I find an error it's extremely time consuming to find the record again to see if it was ultimately successful - I have to log at every single log entry manually. 

This problem happens on a daily basis. This is critical for our business as our team depends on the integration being successful, which means we need to be able to troubleshoot any errors that arise in the logs.