On a new Gentoo install, GDM kept failing to start:

(EE) Cannot open log file "/var/lib/gdm/.local/share/xorg ...log

Found:

The GDM install changed the ownership of /var/lib/gdm/.local/share/applications to gdm.gdm – but it did not change the ownership of the parent directory /var/lib/gdm/.local/share …

Simple solution, chown -R gdm.gdm /var/lib/gdm