Could be a mistranslation but I'll just play it safe.
No, the log doesn't store your progress. Your progress is stored in files that have numbers and end up with LT1.save and the persistent file that saves stuff over all your different saves (like settings).
Your savegame is stored inside the folder called game that's right next where the executable is, inside it there's a folder called saves.
There you have your saves.
Depending on the platform (I'm assuming Mac and Linux work in the same way as Windows) it may also keep a backup folder somewhere else, so if you download a new version of the game without copying over your saves they might still be recognized because it checked the backup.
Save locations list.
WHERE ARE SAVES LOCATED ON ANDROID? Android/data/com.domain.corruptedkingdoms/saves
WHERE ARE SAVES LOCATED ON PC?
PC > Local Disk (usually C:) > Users > USERNAME > AppData > Roaming > RenPy > CorruptedKingdoms-1547378918
WHERE ARE SAVES LOCATED ON LINUX?
~/. renpy/CorruptedKingdoms/
WHERE ARE SAVES LOCATED ON MAC? ~/Library/RenPy/CorruptedKingdoms
NOTE: If you're running Android 11+ then Google has hidden your data folder. To access it you need to connect your phone to your PC with a USB cable or use an app like Solid Explorer or X-Plore to give you access to the folder again.
Hope that helps.