The rapid development of information technologies and artificial intelligence (AI) has transformed nearly every aspect of modern society. Mathematical concepts play a fundamental role in these technological advancements, and logarithms are among the most significant tools used in computational sciences. Logarithms simplify complex calculations, improve algorithm efficiency, and support data processing, machine learning, information theory, and cybersecurity. They provide mathematical foundations for analyzing large datasets, optimizing search algorithms, measuring information content, and training artificial neural networks. This article examines the theoretical basis of logarithms and explores their applications in modern information technologies and artificial intelligence.