It would probably be easier to put it into a script, in which case it should just be as simple as...
cat /dev/null > $LOGFILE
for HOST in host1 host2 host3 host4; do
rsh $HOST <<EOF >> $LOGFILE 2>&1
It should be perfectly fine to indent the rsh commands to make the script more readable, but you MUST place the ending EOF (or whatever tag you decide to use) at the BEGINNING of the line. Otherwise it won't get recognised and end the 'input'