I understand that I can log to the console when using a Firefox extension with the cfx run command.
Is there a way to log to a console after the extens
You need to do 2 things:
enable logging for addons
• in about:config, add a new option "extensions.sdk.console.logLevel" and give it the value "all"
• restart Firefox
in Firefox open the Browser Console:
• Tools -> Web developer -> Browser console
• NOTE: this is different from the usual Web Console used to debug web pages
You should see addons logs there now.
If you run console.log from Add-on code, it send up in the 'Messages' tab of the Error Console window:

Have you tried console.log() with Firebug? (I know you have tried Application.console.log() and Firebug.console.log() already)
I just tried it in the Web Console on FF16 running Firebug 1.10.4 and it seems to work: 
Here are some more examples from the Firebug Wiki itself: FirebugWiki Console API
Additionally, you can write messages of different types in the console, such as: console.debug(), console.info(), console.warn(), console.error()
TLDR:
about:config url and create key extensions.sdk.console.logLevel with value allcfx or its successor jpm creates this configuration key automatically in development firefox profile.
From logging documentation:
extensions.sdk.console.logLevel: if set, this determines the logging level for all installed SDK-based add-ons.
extensions.extensionID.sdk.console.logLevel, where extensionID is an add-on's Program ID. If set, this determines the logging level for the specified add-on. This overrides the global preference if both preferences are set.