I found that my Logos 6 desktop does not connect to logos.com.
I tried both on Windows and on Mac systems and got the same result: the dialogue "Sign in with your Logos.com account" outputs error "Unable to connect to Logos.com".
It seems that thre might be some problem with logos.com server.
Logos 6 is no longer supported nor will it connect to FL servers. You can still use L6 offline. I recommend downloading the free L8 engine. That will resolve your connection issues
Logging Tool
Logos Wiki advises versions prior to 7.18 can not log into Logos servers from 10th December 2018.
https://wiki.logos.com/Unsupported_Obsolete_Applications
You should be able to go to Logos website and download the latest version of the base software for free to enable you to connect again to the Logos server If your system meets the minimum requirements.
Be aware though your notes will be converted over to the new format and the interface will change, most noticeably the home page so you might want to watch some of the training videos available before you upgrade your installation just to familiarise yourself with the changes.