Output shell script to log file - unix

Question

I have a scheduled unix script that I want to log the output of. I am unable to edit the cron file due to the user interface restrictions, and I am unable to add >> logfile to the command. Is there something I can add within the script itself to send the output to a log?
{
printf poo
#Do not change
PRINTF=/usr/bin/printf
MSMTP=/usr/local/bin/msmtp
MSMTPCONF=/var/etc/msmtp.conf
#Can be changed
FROM="nas4free#usinfosec.com"
TO="dpatino#usinfosec.com"
MDIR="CaseData"
SUBJECT="$MDIR Backup Report"
} > /mnt/support/logs/$SUBJECT.log
#BODY="$(cat /mnt/support/logs/test.log)"
#$PRINTF "From:$FROM\nTo:$TO\nSubject:$SUBJECT\n\n$BODY" | $MSMTP --file=$MSMTPCONF -t

Answer 1

Try
#!/bin/bash
exec > /tmp/myLog.log 2>&1
set -x
The log shows:
+ echo 'Hello World!'
Hello World!

Answer 2

One way is to wrap your script in braces and redirect the output as shown below:
#!/bin/bash
{
# script contents here
echo running script
} > logfile

Answer 3

Append following line in starting of your script
log_file_path="/tmp/output.log"
log() { while IFS='' read -r line; do echo "$line" >> "$log_file_path"; done; };
exec > >(tee >(log)) 2>&1
Your script with modification
PRINTF=/usr/bin/printf
MSMTP=/usr/local/bin/msmtp
MSMTPCONF=/var/etc/msmtp.conf
FROM="nas4free#usinfosec.com"
TO="dpatino#usinfosec.com"
MDIR="CaseData"
SUBJECT="$MDIR Backup Report"
{
printf poo
} > /mnt/support/logs/$SUBJECT.log
#BODY="$(cat /mnt/support/logs/test.log)"
#$PRINTF "From:$FROM\nTo:$TO\nSubject:$SUBJECT\n\n$BODY" | $MSMTP --file=$MSMTPCONF -t