Just because "log" is the most relevant keyword Google has found on your site doesn't mean your site will rank well for "log" or "log in". It just means that keyword is very prevalent in your site. And that's not a bad thing. Your pages can be relevant for lots of keywords, some of which will be mundane like "log" thanks to common links in your header or footer, and others because that is what the page is about.
What you're seeing is common to most websites. You shouldn't make any changes to your site just to make that list show the keywords you want to rank well for. The two are only loosely linked.
Just focus on improving your semantic markup as that clearly is an area for improvement for your site. Then make sure you get quality links, etc. But definitely go making changes to make that list happy. That's just a clear case of focusing on the wrong thing.