output both stderr and stderr on consle and store them in a file at same time

may I output both stdout and stderr on console screen and store one of them into a log file?

I write a test shell script:

#!/bin/sh

echo OUT! >&1
echo ERR! >&2

I can output both of them on screen just by run the script:

$./test 
OUT!
ERR!

I can output stderr and catch stdout into log file by:

$./test | tee 1>log
ERR!

$cat log 
OUT!

I can output nothing but catch all stdout and stderro into log file by:

$./test 2>&1| tee 1>log

$cat log 
OUT!
ERR!

I can output both of stdout and stderr and catch all of them into a log file by:

$./test 2>&1 | tee log
OUT!
ERR!

$cat log 
OUT!
ERR!

I can output both can catch stdout into log file by:

$./test | tee 2>&1 log
ERR!
OUT!

$cat log 
OUT!

My questions are:

1. how to just output stdout and catch stderr into file?(I tried ./test|tee 2>log, but doesn't work)


2. how to just output both and catch stderr into file?