You'll need to configure your Synology to allow you to SSH into it, guides to this shouldn't be difficult to find. Do make sure you don't accidentally allow the SSH port to be available across the Internet.
I've changed the settings the Package Manager used so this may not be correct, if so just let me know
The logfile should be saved at
You can copy this to one of your user-readable folders, this command should probably work, replacing "foldername" with one of your shared folders:
cp /usr/local/syncthing/var/syncthing.log '/volume1/foldername'