In this paper, we present the design and architecture of REI, a novel system for indexing log data for regular expression queries. Our main contribution is an $n$-gram-based indexing strategy and an efficient storage mechanism that results in a speedup of up to 14x compared to state-of-the-art regex processing engines that do not use indexing, using only 2.1% of extra space. We perform a detailed study that analyzes the space usage of the index and the improvement in workload execution time, uncovering interesting insights. Specifically, we show that even an optimized implementation of strategies such as inverted indexing, which are widely used in text processing libraries, may lead to suboptimal performance for regex indexing on log analysis tasks. Overall, the REI approach presented in this paper provides a significant boost when evaluating regular expression queries on log data. REI is also modular and can work with existing regular expression packages, making it easy to deploy in a variety of settings. The code of REI is available at https://github.com/mush-zhang/REI-Regular-Expression-Indexing.