The XDG spec is not very clear on where log files should go. The best fit seems to be under $XDG_CACHE_HOME which is for "user specific non-essential data files".
Looking at my local system, there are application logs under all of the $XDG_*_HOME locations:
$ ls .cache/*/*log
.cache/shotwell/shotwell.log
.cache/simple-scan/simple-scan.log
.cache/virt-manager/virt-clone.log
$ ls .local/share/*/*log
.local/share/FasterThanLight/FTL.log
.local/share/xorg/Xorg.0.log
$ ls .config/*/*log
.config/VirtualBox/selectorwindow.log
.config/VirtualBox/VBoxSVC.log
Using $XDG_CONFIG_HOME seems wrong, but can see using the (more likely to persist through reboots) $XDG_DATA_HOME specifically for applications like Xorg which may break hard enough to require a system restart and want to be able to preserve error logs in these scenarios.