I have developed an Android application that performs some network activity. Inside the classes I have implemented the log messages like verbose, debug, info, warn and error
Logcat has a small + button that lets you create a filter. Use the TAG you used in your app and specify it in the filter. That creates a new tab that shows only your apps messages. While you're at it, I recommend creating a filter for the AndroidRuntime tag. All your exceptions get dumped in there and you can find them very easy.
Of course you can filter these messages more by category (verbose, warn, error, ..) by using the buttons in the same bar.