environment variables in redirectors, perfecting my log scripts

The following command, run either from cron or the prompt, does not make the expected log. I expect to get a log with the date as the file name. I a开发者_StackOverflowm able to do this in Windows by using %date% and it makes the date the filename. What am I doing wrong here.

/root/backup_scripts/new_scripts/test.sh>/root/backup_scripts/new_scripts/$date.log 2>&1

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/root/backup_scripts/new_scripts/test.sh > "/root/backup_scripts/new_scripts/$(date).log" 2>&1