Well, I use console.log() for logging. Simple as that. That method writes a string to stdout and you can see it in a console. Since it is just a console, you can even color and style the font for better readability.

While my application is running, I additionally write rapidly repeating events (e.g. someone calls my API or makes a web request) to a file. I really recommend some kind of streaming log-rotation mechanism for that.

Since errors and warnings are very important I recommend logging them to a separate place. I like databases, but you might just use another file or even console.warn() or console.error() which write to stderr (which is equal to stdout by default), as well.