On the wide variety of Unix and "Unix-like" systems that exist, log files are structured in diverse ways. But in general, if the system is configured to have cron log it's activities, cron tends to do some basic logging of the tasks it runs. So in additional to your detailed/trace output, I'd expect cron to do some basic logging about the task you are having it run.
In your situation, you've described what's happening on your system, but NOT how you want things to work differently. What is that you want to happen differently that it is happening?