Yes, it is possible to log, in fact is it already logged by the kernel as standard.
And you can access it from both the dmesg and the system log.
cat /var/log/syslog. Be aware this will also send you email on reboot and startup of the system.
Wish you good luck and have fun tweaking it to perfectly suit your needs.
# Home crafted udev usb mail rule
ACTION=="add", SUBSYSTEMS=="usb", RUN+="/home/yourusername/sendudev.sh"
ACTION=="remove", SUBSYSTEMS=="usb", RUN+="/home/yourusername/sendudev.sh"
text="$(/bin/dmesg | /bin/grep -i 'usb' | tac)"
/bin/echo -e "Date:$(date)\n\nudev env and reversed log is embed.\n\n$(env)\n\n$text" | /usr/bin/mail -s 'Mail from udev.' -t root@localhost
sudo chmod +x /home/yourusername/sendudev.sh