You can pipe the script's output through a loop that prefixes the current date and time:
./script.sh | while IFS= read -r line; do echoprintf '%s %s\n' "$(date)" $line";"$line"; done >>/var/log/logfile If you'll be using this a lot, it's easy to make a bash function to handle the loop:
adddate() { while IFS= read -r line; do printf "%s'%s %s\n"%s\n' "$(date)" "$line"; done } ./thisscript.sh | adddate >>/var/log/logfile ./thatscript.sh | adddate >>/var/log/logfile ./theotherscript.sh | adddate >>/var/log/logfile