We are now testing our application with a few friends. Sometimes there are some errors which don't throw an exception. So I don't really know whats the problem was. So I thought it would be a good idea to implement a menu item which allows to send the logcat output to a e-mail address, so that we can examine the log.
Unfortunately I didn't find a hint in the Internet how to extract the logcat from a phone. How to send an email shouldn't be the problem.
Look at android-log-collector, as it does what you are trying to do.
It is not possible to collect arbitrary LogCat data as of Android 4.1. There was never a documented and supported way of doing that, and the undocumented/unsupported way was locked down in Jelly Bean. For your own crashes, you are better served using a crash logging library, like ACRA.