To get the logs of a GNOME Extension, you must first find out its UUID (which will look like name@author.address.com). Look for it in the output of:

ls $HOME/.local/share/gnome-shell/extensions

You can then get its logs with the command:

journalctl -f -o cat GNOME_SHELL_EXTENSION_UUID=name@author.address.com

Some logs from gjs (GNOME JavaScript) may also interest you, so you can checkout the output of:

journalctl -f -o cat /usr/bin/gjs

Sources: [1] [2]