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Data science is revolutionizing academia and industry, creating a high demand for a workforce fluent in this field. While the availability of data science courses has increased recently, few curricula rigorously build on mathematical logic. The LogicDS Project addresses this gap by engaging high school students from rural communities in an online data science course integrating mathematics, statistics, and programming into a unified framework based on logic and reasoning. A one-week course, consisting of six lessons, was developed and 110 participants were recruited. Pre- and post-intervention data, along with students' LMS activity logs, were collected to analyze engagement. Results indicate that the Logic-Based framework effectively engages students from diverse backgrounds, with participants finding the course valuable for learning data science skills. Notably, entropy analysis of student activity logs correlated with other mixed methods analyses, providing insights into engaging K-12 students in data science education.