Where to find logs for "Low Disk Space"

After seeing the window popping up that shows

Low disk space | The volume "Filesystem root" has only X MB disk space remaining.

Where can we find the log file that logs this low disk space warning?

If it is not logged by default, how can we log it so that we can record down some details like the timestamp of this event?

Output of cat /var/log/syslog | grep -i disk:

Sep  5 09:10:11 foo systemd[1]: dev-disk-by\x2duuid-4d64b496\x2d3e08\x2d497f\x2d9e51\x2d40db25c7710b.device: Job dev-disk-by\x2duuid-4d64b496\x2d3e08\x2d497f\x2d9e51\x2d40db25c7710b.device/start timed out.
Sep  5 09:10:11 foo systemd[1]: Timed out waiting for device dev-disk-by\x2duuid-4d64b496\x2d3e08\x2d497f\x2d9e51\x2d40db25c7710b.device.
Sep  5 09:10:11 foo systemd[1]: Dependency failed for File System Check on /dev/disk/by-uuid/4d64b496-3e08-497f-9e51-40db25c7710b.
Sep  5 09:10:11 foo systemd[1]: systemd-fsck@dev-disk-by\x2duuid-4d64b496\x2d3e08\x2d497f\x2d9e51\x2d40db25c7710b.service: Job systemd-fsck@dev-disk-by\x2duuid-4d64b496\x2d3e08\x2d497f\x2d9e51\x2d40db25c7710b.service/start failed with result 'dependency'.
Sep  5 09:10:11 foo systemd[1]: dev-disk-by\x2duuid-4d64b496\x2d3e08\x2d497f\x2d9e51\x2d40db25c7710b.device: Job dev-disk-by\x2duuid-4d64b496\x2d3e08\x2d497f\x2d9e51\x2d40db25c7710b.device/start failed with result 'timeout'.
Sep  5 09:34:55 foo systemd[1]: dev-disk-by\x2duuid-4d64b496\x2d3e08\x2d497f\x2d9e51\x2d40db25c7710b.device: Job dev-disk-by\x2duuid-4d64b496\x2d3e08\x2d497f\x2d9e51\x2d40db25c7710b.device/start timed out.
Sep  5 09:34:55 foo systemd[1]: Timed out waiting for device dev-disk-by\x2duuid-4d64b496\x2d3e08\x2d497f\x2d9e51\x2d40db25c7710b.device.
Sep  5 09:34:55 foo systemd[1]: Dependency failed for File System Check on /dev/disk/by-uuid/4d64b496-3e08-497f-9e51-40db25c7710b.
Sep  5 09:34:55 foo systemd[1]: systemd-fsck@dev-disk-by\x2duuid-4d64b496\x2d3e08\x2d497f\x2d9e51\x2d40db25c7710b.service: Job systemd-fsck@dev-disk-by\x2duuid-4d64b496\x2d3e08\x2d497f\x2d9e51\x2d40db25c7710b.service/start failed with result 'dependency'.
Sep  5 09:34:55 foo systemd[1]: dev-disk-by\x2duuid-4d64b496\x2d3e08\x2d497f\x2d9e51\x2d40db25c7710b.device: Job dev-disk-by\x2duuid-4d64b496\x2d3e08\x2d497f\x2d9e51\x2d40db25c7710b.device/start failed with result 'timeout'.
Sep  5 10:03:12 foo systemd[1]: dev-disk-by\x2duuid-4d64b496\x2d3e08\x2d497f\x2d9e51\x2d40db25c7710b.device: Job dev-disk-by\x2duuid-4d64b496\x2d3e08\x2d497f\x2d9e51\x2d40db25c7710b.device/start timed out.
Sep  5 10:03:12 foo systemd[1]: Timed out waiting for device dev-disk-by\x2duuid-4d64b496\x2d3e08\x2d497f\x2d9e51\x2d40db25c7710b.device.
Sep  5 10:03:12 foo systemd[1]: Dependency failed for File System Check on /dev/disk/by-uuid/4d64b496-3e08-497f-9e51-40db25c7710b.
Sep  5 10:03:12 foo systemd[1]: systemd-fsck@dev-disk-by\x2duuid-4d64b496\x2d3e08\x2d497f\x2d9e51\x2d40db25c7710b.service: Job systemd-fsck@dev-disk-by\x2duuid-4d64b496\x2d3e08\x2d497f\x2d9e51\x2d40db25c7710b.service/start failed with result 'dependency'.
Sep  5 10:03:12 foo systemd[1]: dev-disk-by\x2duuid-4d64b496\x2d3e08\x2d497f\x2d9e51\x2d40db25c7710b.device: Job dev-disk-by\x2duuid-4d64b496\x2d3e08\x2d497f\x2d9e51\x2d40db25c7710b.device/start failed with result 'timeout'.

This dialog is generated by desktop settings daemons. Depending on the flavor the list includes:

• cinnamon-settings-daemon
• gnome-settings-daemon
• mate-settings-daemon (your case, based on comments)
• ukui-settings-daemon

For MATE log monitoring shows that this low-space warning is not logged. You can check space distribution using MATE Disk Usage Analyzer (mate-disk-usage-analyzer) or via CLI - using ncdu. Then you can move/remove huge files in your home directory.

Cleaning old kernels, kernel headers and package cache may be useful - try commands below:

sudo apt-get clean
sudo apt-get autoremove --purge