Usually, console programs have the feature to limit size of a logging-file. I.e. logging-file have max size is 500 Kbytes for example. This size is more than enough to keep the information about CCleaner`s work in the last week. CCleaner should just append new records to the end of a logging-file. So, I can see a logging-file in any time.
Actually, personally for me, I don't need even a logging-file. If something goes wrong, I see it during a work with the Windows, applications, etc.