I know this doesn't "fix" the issue of optimizing the logmein123.com page's search rankings, but what I have done was to put the code (which logmein makes available to everyone) into my own webpage.
This solved two different problems for me, 1) I was able to create a unique, more easily identifiable domain name for people to find (you can then control the search engine optimization and related ranking), and 2) it helps me mask the technology just enough to make it look like we have created a value added resource, not just used someone else's technology.
For those who are in a corporate environment or use a help desk system, you can simply embed the code in one of your own (existing) web pages, or within a subdomain. i.e.. If your corporate site is www.mycorp.com, you can create a direct login page with a subdomain like login.mycorp.com.
This has worked great for us...