Log analytics are a critical part of the operational management in today's IT services. However, the growing software complexity and volume of logs make it increasingly challenging to mine useful insights from logs for problem diagnosis. In this paper, we propose a novel technique, Priolog, that can narrow down the volume of logs into a small set of important and most relevant logs. Priolog uses a combination of log template temporal analysis, log template frequency analysis, and word frequency analysis, which complement each other to generate an accurately ranked list of important logs.We have implemented this technique and applied to the problem diagnosis task of the popular OpenStack platform. Our evaluation indicates that Priolog can effectively find the important logs that hold direct hints to the failure cause in several scenarios. We demonstrate the concepts, design, and evaluation results using actual logs.