Hi and welcome to LinuxQuestions!
The syntax is correct and indeed it should work, provided the crontab's owner has permissions to write in /home. You can check three things:
1. see if the cron log actually reports the execution of your job. Usually it is /var/log/cron and you should see an entry like this:
Jul 1 10:11:02 hostname crond: (username) CMD (/var/test.sh > /home/log)
2. since you redirect only the standard output to /home/log, the standard error (if any) would be sent to the crontab's owner mailbox. You can check it from a terminal with the mail
3. see if the crontab's owner has permissions to run cron jobs. You can check the /etc/cron.deny and /etc/cron.allow files.
In any case, who is the crontab's owner? A regular user or root? And how did you edit the crontab (using the command crontab -e or something else)?
Edit: please note that the mkdir entry is wrong instead. It should be something like:
0 13 * * * cd /home && mkdir asdasd
or mkdir /home/asdasd.