All you need to do is paste the following textblock under the shebang in your script! That's all!
You'll find logfiles in /root/ [only works for root, change line 3 if running as a different user]
# All in One Bash Logger | v0.46 | 20171018 | 20171107 | Nk
logname=`basename "$0"` # The name of this script
now="$(date +"%Y-%m-%d_%H-%M-%S")" # The current timestamp
logdir="/root/logs/$logname" # Don't store anything else than logs in here!
logfile="$logdir/$now" # The new logfile
mkdir -p $logdir # Touch the dir
touch $logfile # Touch the file
rm $logdir/latest-log 2> /dev/null # Remove the old latest-log symlink
ln -s $logfile $logdir/latest-log # Recreate the symlink
( cd $logdir && rm `ls -t | awk 'NR>43'` ) 2> /dev/null # Delete all logs older than the newest 42
exec > >(tee -ia $logfile) # Log one output to logfile
exec 2> >(tee -ia $logfile >&2) # Log the other output to logfile
echo && echo "Starting $logname script on $now..." && echo # Write heading to logfile