Is there a log for copied files?

Not really. If the file has been copied using a command line, then chances are that you can find the information in $HOME/.bash_history. But then again, not necessarily, bash history is not a logfile.

Kernel logs contain information about attaching usb devices, so they can be monitored for inserting a pen drive, for example.

If you really need to do such a thing, then you have to set up monitoring the selected devices by yourself.