Where would the response go?
You might try looking at mod_security, which has a number of useful features that would be of use... is your goal security/forensics, or performance analysis?
For performance analysis, I've found it more useful in the past to create a more comprehensive logging format that captures things like response-code, response Location header (for tracking redirects), selected request headers, timing information, etc.
If https is not in use, then you might be better served by something driven by packet-capture. I know that Oracle Real User Information (?) (RUI) works using that principle. For more casual diagnostic sessions, I've often gotten away with the following tcpdump:
tcpdump -s0 -A -p -nn tcp and port 80
That's enough to get the full requests (and responses), it is a little messy, but the data is all there. You can clean it up a bit with a script, such as the following (tcpdump-http-headers-only) -- its not perfect (particularly on a busy server where things get harder to track).
#!/bin/bash
#
# Pass in the output of 'tcpdump -s0 -A ...' to this and it will
# output only the HTTP request headers and response headers.
#
# Cameron Kerr <cameron.kerr.nz@gmail.com>
# 2013-02-14
#
grep --line-buffered -o \
-e $'GET .*\r' \
-e $'POST .*\r' \
-e $'^[A-Z][A-Za-z0-9_-]*: .*\r' \
-e $'HTTP/1.1 .*\r' \
-e $'^\r$' \
| sed --unbuffered -e 's,\r$,,'
Alternatively, you might like to capture them (perhaps in conjunction with the -W, -C or -G options) for later analysis. This can, depending on the cipher used, also work with https connections if the key is provided (useful for Wireshark)