There is a method in the Phone to read the logs. However I don't recommend you to implement such a system. It is more complicated than it may seem.
Also, how will you send the logs once your application crashes?
Applications uploaded to the market include this feature automatically. When there is a crash, the user is asked whether they want to send a crash report.
so how does waze, implemented their mechanism
when there is a problem, I am getting - "please help us to improve our app, by sending the log" and I have "yes"/"no" to send the file or such
You can catch the exceptions with Try / Catch and then send the information. As I wrote before, such a service is already implemented in the OS for market applications.