Ask HN: How do you manage server logs?

I'm at the "we should start to think about logging" phase of a new project, and the recent "How I store server logs in Google Spreadsheets" post (https://news.ycombinator.com/item?id=8317868) has me wondering:

How do you manage your server logs?

Which libraries or technologies work well? Which don't? Which techniques do you have experience with? What do you like about them? What scales well with multiple machines?

How long do you store your logs? Do you use server logs for performance and usability analytics, or do you record that information elsewhere?