In this paper we introduce the Learning Management System (LMS) eLogika that has been developed in our department for teaching mathematical logic. There were many reasons that led us to the decision to develop such a system, including inter alia a great amount of students enrolled for the courses on logic. Yet the most important reason was a specific character of logic education. As a result, the eLogika system is a web application that provides didactic material for courses on mathematical logic. Its main goal is an automatic test generation and computer-aided test evaluation based on a large database of logic tasks. The system makes it possible to adjust the level of particular tests according to students' knowledge level. To this end we developed a feedback module that makes use of statistics and data mining methods. The system can generate a large number of training as well as exam test variants for each common thematic topic. At the same time it provides effective semi-automatic methods of test rating and evaluation. In the paper we describe particular modules of eLogika with the focus on the modules of data mining and statistics.