Log parsing transforms log messages into structured formats, serving as the prerequisite step for various log analysis tasks. Although a variety of log parsing approaches have been proposed, their performance on complicated log data remains compromised due to the use of human-crafted rules or learning-based models with limited training data. The recent emergence of powerful large language models (LLMs) demonstrates their vast pre-trained knowledge related to code and logging, making it promising to apply LLMs for log parsing. However, their lack of specialized log parsing capabilities currently hinders their accuracy in parsing. Moreover, the inherent inconsistent answers, as well as the substantial overhead, prevent the practical adoption of LLM-based log parsing. To address these challenges, we propose LILAC, the first practical log parsing framework using LLMs with adaptive parsing cache. To facilitate accurate and robust log parsing, LILAC leverages the in-context learning (ICL) capability of the LLM by performing a hierarchical candidate sampling algorithm and selecting high-quality demonstrations. Furthermore, LILAC incorporates a novel component, an adaptive parsing cache, to store and refine the templates generated by the LLM. It helps mitigate LLM's inefficiency issue by enabling rapid retrieval of previously processed log templates. In this process, LILAC adaptively updates the templates within the parsing cache to ensure the consistency of parsed results. The extensive evaluation on public large-scale datasets shows that LILAC outperforms state-of-the-art methods by 69.5% in terms of the average F1 score of template accuracy. In addition, LILAC reduces the query times to LLMs by several orders of magnitude, achieving a comparable efficiency to the fastest baseline.