You may google "Logging as a service", it's 2017 ftw, there is X as a service today for everything. However, I have no idea why you even need it?
What is a problem with logging system your app and web server gives you, e.g. just default log files? For app logs in PHP I use Laravel's event and exception system, whenever I want to know something critical, like failed admin logins, delete operations executed from admin panel, any app exceptions and errors I instantly receive an email. I suppose it's easy to configure and set up in any app/framework you use depending on your stack. Usually I also have a Logs section in admin panel with a simple datatable UI, it's not harder to implement either.
Talking about the server logs I have nginx default error logs and access logs. Whenever I need to get some info from those files I just write a simple script. You even may parse those files from app itself and display all the results you need in the UI you want.
There is also nothing to scale, every server/instance should store logs in files. nginx #1 has own logs, nginx #2 owns, etc. Usually those simple log files are gzipped time-to-time and old ones removed with a simple script/cron.
Fact is it's something simple and specific logic always will depend on your specific use-cases, that's why I see no other option but writing own bash scripts/cron tasks/app scripts on the top of the default log files yourself