It's an old post, with already good answers, but I add my two bits. I don't like to use console.log, I'd rather use a logger that logs to the console, or wherever I want, so I have a module defining a log function a bit like this one
function log(...args) {
console.log(...args);
chrome.extension.getBackgroundPage().console.log(...args);
}
When I call log("this is my log") it will write the message both in the popup console and the background console.
The advantage is to be able to change the behaviour of the logs without having to change the code (like disabling logs for production, etc...)