I'm pretty sure you can't write to couch.log from a view, it's a sandboxed system.
Getting a record of connections to the server is possible though. Here's a dump from my couch.log, with an HTTP error in there:
/
[Sat, 13 Sep 2014 08:18:57 GMT] [info] [<0.160.0>] Opening index for db: test idx: _design/ivet sig: "f6b64ef8593e23cac644c13b895b7607"
[Sat, 13 Sep 2014 08:18:57 GMT] [info] [<0.121.0>] 127.0.0.1 - - GET /test/_design/ivet/_view/medicationWHP/foobar?include_docs=true 200
[Sat, 13 Sep 2014 08:18:57 GMT] [info] [<0.121.0>] 127.0.0.1 - - GET /test/_design/ivet/_view/medicationWHP/foobar?include_docs=true 500
[Sat, 13 Sep 2014 08:18:57 GMT] [error] [<0.121.0>] httpd 500 error response:
{"error":"json_encode","reason":"{bad_term,{key,null}}"}
[Sat, 13 Sep 2014 08:19:05 GMT] [info] [<0.36.0>] Apache CouchDB has started on http://127.0.0.1:5984/
You can see it has the VERB PATH CODE format for each line, so you can filter that for whatever you need. (Unauthorized is 401) You can also access the log through /_log. Details on that are here:
http://docs.couchdb.org/en/latest/api/server/common.html#log
To get all that information, you'll need to have the log level set to info. You can do this at the config screen in futon.
To do it server-side, you'd probably need to use node.js or something like that. Just have it consume the /_log endpoint, and filter each line by the HTTP response code.