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redirect all error output to a file, &» /tmp/whatever.log isn't support by sh, so use command below instead:
12 * * * * /opt/homebrew/bin/brew update >> /tmp/brew.log 2>&1 30 */3 * * * /opt/homebrew/bin/rsync -av -O --delete /Users/dcai/Dropbox /backups/machinefolder/ >> /tmp/backup.log 2>&1
Adding -O / –omit-dir-times to avoid setting times on directories (which causes errors in ntfs)