Microlog is recommended in this article:
Microlog is a small, yet powerful logging library for mobile devices based on the Log4j API. Supports Java ME (J2ME) and Android. Logs to device, to PC or to servers online. Used in all phases from development on emulator/device to outdoor field-testing.
Microlog is licensed under The Apache Software License, Version 2.0 so it is possible to link and distribute commercial software with this library.