Nah...there's boatloads of Browser sniffers out there. It is totally common to detect the users browser without telling them. Hell, that JS I wrote for you did it...it just didn't log it. But, in order for you to capture it to a file, you have now moved into CGI's. This is really easy and totally doable, but you might just want to talk to your web-host and see if thier logs have that info (chances are they do unless they are doing no logging at all for you).

If you want to do it yourself. PERL makes this really simple...

#!/path/to/perl

open (LOG, ">>/path/to/your/file.log");
print LOG "$ENV{USER_AGENT}\n";
close (LOG);
exit;

(Save this to a file of course...this has to run on the server)

You will need to probably execute this with an SSI command wich means that you need to be able to use Server Side Includes on your server. You can also use and image source to fire it. Do a Google search for browser detection, I'm sure you;ll find a solution that is good for your situation.