Data science is revolutionizing academia and industry, and there is a strong demand for a workforce fluent in data science. The availability of such courses has increased substantially in recent years. However, there are few rigorous curricula built on mathematical logic as the foundation of data science. The LogicDS Project aims to engage high school students from rural communities in an online data science course that integrates mathematics, statistics, and programming concepts into a single unified framework based on logic and reasoning. We developed a one-week data science course consisting of six lessons (a sampling of the full course offering) and recruited 110 participants. We collected pre- and post-intervention data and students' LMS activity log data to analyze their engagement. Results indicate that our Logic-Based framework for data science education effectively engages students from a variety of backgrounds; it was shown that they perceived the course as valuable for learning data science skills/concepts. Notably, our focus on the entropy analysis of student activity logs correlated to our other mixed methods analyses. This study provided insights for engaging K-12 students learning data science.