You can send logs to influx no problem. telegraf (The collecting agent from the Influx team) has a logparser input plugin which can parse logs via patterns, it understands Apache logs by default. The catch is that there is no full text search. It's still useful as you can quickly find logs to view by searching fields that have been parsed.
That's really funny. My employer is one of the larger users of influxdb I'm aware of, so much so we had to write some software[1] (we did synthetic benchmarks of 500k metrics per second and it held up just fine) to overcome scaling limitations and I didn't know this. Thanks for the cluebat! TIL!
Namely InfluxDB and friends