I need to gather different kinds of logs:
Which tool is more appropriate to do that?
My infrastructure is composed by Kubernetes orchestring docker machines installed in Microsoft azure (using ACS engine and RPM).
You can use Application Insights. App Insights can be used to store logging/monitoring for Docker Apps easily
See this blog for understanding how Application insights support microservices and containers.
You can easily pull the microsoft/applicationinsigthts
image from docker hub and start using it.
Here the pricing calculator for application insights (Microsoft will give 5GB of Storage for logs free for every month)