How do I write outputs to the Log in Android?

Look into android.util.Log. It lets you write to the log with various log levels, and you can specify different tags to group the output. For example

Log.w("myApp", "no network");

will output a warning with the tag myApp and the message no network.

The Tag is just used to easily find your output, because the Output of LogCat can be sometimes very long. You can define somewhere in your class:

private static final String TAG = "myApp";

and use it when debugging

Log.v(TAG, "did something");

You can apply as well a Filter to only search for the tag.