Background logcat with filter
You could use a logcat app that writes the log to your sd card. Depending on the Android version on your device, that might require root permissions: latest with Android 4.2 "normal apps" only can see their own log. Checkout my list for Logcat &amp; Co. for possible candidates. I know that e.g. aLogCat has the feature I've described. It also has filtering capabilities. Same applies to several other candidates of the linked list.