The log files are stored in the Library folder, which is hidden by default. You can enable full path in Finder by checking View > Show Path Bar. The full path should be something like this:
/Users/<USERNAME>/Library/Application Support/ ManyCam/ManyCam/Logs
How to retrieve the log files from ManyCam:
1. Download and install the latest version of ManyCam,
2. Start ManyCam and reproduce the issue,
3. Close ManyCam: ManyCam menu > Quit ManyCam,
4. Navigate to the following location (Go > Go to Folder):
~/Library/Application Support/ ManyCam/ManyCam/Logs
Collect all the files from this location.