Change it to this:
www.mydomain\?campaign=google
In your logs you should now see it turn up in the query string field.
Having said that you will never end up with the same number as in the Google reporting. When you track (anlything) online using two different tracking mechanisms (yours and Googles) you will always end up with two different numbers. On top of that comes the filtering applied to each system making it impossible to get to the same numbers.
You will get a more precise result if you use a tracker instead of logfiles for this as that will count all cached page views too.