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xLog
Android logger, pretty, powerful and flexible, log to everywhere, save to file, all you want is here. (by elvishew)
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InfluxDB
Power Real-Time Data Analytics at Scale. Get real-time insights from all types of time series data with InfluxDB. Ingest, query, and analyze billions of data points in real-time with unbounded cardinality.
There are also some logging libraries which try to make logcat but "better" and can write to disk, I haven't tried these myself https://github.com/elvishew/xLog
NOTE:
The number of mentions on this list indicates mentions on common posts plus user suggested alternatives.
Hence, a higher number means a more popular project.