The canonical way to do this is with fs.watchFile.
Alternatively, you could just use the node-tail module, which uses fs.watchFile internally and has already done the work for you. Here is an example of using it straight from the documentation:
Tail = require('tail').Tail;
tail = new Tail("fileToTail");
tail.on("line", function(data) {
console.log(data);
});