Hello! I just came across your question. Here is a recent Moz post of mine on how to use log files for technical SEO -- the example screenshots are from the SEO dashboard in Logz.io. Feel free to sign-up at our site and test our public beta to see if it works for you.

Disclosure: Yes, I work for the company.

Thanks for the question! I'm sure the community would love to help, but the question is a little vague. If it's possible, could you tell us your brand name and website URL? There are numerous issues that could be involved, so it'd be hard to help unless we can get more details.

I'm getting a sense that you may be suffering from a duplicate-content issue. But first, let me explain.

Duplicate content is not exactly a "penalty" (most of the time). Here's what usually happens: Google sees that two pages of your website are essentially the same, and the search engine does not want to have redundant search results. So, it will usually make a decision to include one of the two in search results (and not the other).

I have to ask: Why do you essentially have two home pages? Is it only to attempt to have the home page rank for two different sets of keywords? If so, then I'm guessing that your two home pages are essentially the same except for some minor keyword variations. It's likely that Google changed its mind (for whatever reason) on which to include and which to ignore.

I'd keep the example.com page as the main home page and 301 (permanent) redirect example.com/index/ to that page. The next time that Google indexes your website, it should see the change and then the SERP should reflect the change shortly thereafter.

In general, it is best to do all of the changes (the domain, branding, and URL work) all on a development site (that is blocking search engines) and then release the new site only after everything is complete. If you space it out piecemeal over too long of a time, it will look very messy both to Google and your website visitors. Releasing a new, properly-migrated website all at once is the quickest way to mitigate any "SEO" damage and return to your prior traffic and ranking levels quickly.

However, it'd be impossible for me or anyone to be more specific without doing a deep dive into your site. However, there are some general best practices for site migrations. For reference on all of these, I'd take a look at what Moz did when migrating from seomoz.org to moz.com. Moz has a webinar and associated blog post that goes through all of the technical details and results. Aleyda Solis also has a good migration checklist with links to more resources here.

I hope that helps -- good luck!

Just a couple thoughts off the top of my head:

1. Double-check all technical international SEO issues and ensure that the robots.txt file is not mistakenly blocking any desired pages.

2. Make sure that you have a separate Google Webmaster Tools setup for each root domain / subdomain / subdirectory (however you have set up the international sites) and have submitted an individual XML sitemap for each one. Also make sure that the geographical targeting in each GWT setup is set to the desired country.

3. If Google is only indexing a small percentage of a site's pages, it is often because Google is thinking (accurately or not) that a site has duplicate content. "Duplicate content" is not a penalty per se -- it is when Google, for example, sees two pages that are very similar and then indexes only one of them so as to not provide redundant pages in search results.

Example: Say that you have an e-commerce product that has ten variations (such as color). The content of each variation page would often be very similar except, for, the listed color. In the case, you would want to use a rel=canonical tag on all variation pages that points to the main page for that product. (In other words, you don't want all of those pages to be indexed, and Google often would not index them anyway.)

Most likely, I would use a tool such as Moz or any other SEO software to crawl the site and see if any duplicate-content issues are present. Once these are addressed (if the problem exists), then Google will likely crawl and index your sites more thoroughly and accurately.

I hope this helps -- good luck!