Store the output of date and watch command to a file
Solution 1:
In order to do what you are looking for, a simple script (as @Ignacio pointed out) should do the trick:
while true
do
echo "$(date '+TIME:%H:%M:%S') $(ps aux | grep "pattern" | wc -l)" | tee -a logfile
sleep 2
done
I use tee instead of >> so that you can see the output on your terminal as well as capture it in your log.
Solution 2:
This can easily be done using watch too without using any scripts.
watch -t -n 10 "(date '+TIME:%H:%M:%S' ; ps aux | grep "pattern" | wc -l) | tee -a logfile"