The growing complexity of log data in modern software systems has prompted the use of Large Language Models (LLMs) for automated log analysis. Current approaches typically rely on direct supervised fine-tuning (SFT) on log-label pairs. However, this exacerbates the domain discrepancy between general-purpose LLMs and specialized log data, causing overfitting. Furthermore, SFT's imbalanced loss computation often allows lengthy contexts to overwhelm critical, concise details in model answers, leading to hallucinations. To address these limitations, we propose R-Log, a novel reasoning-based paradigm that mirrors the structured, step-by-step analytical process of human engineers. This approach enhances generalizability by learning the underlying rules behind conclusions. We further employ Reinforcement Learning (RL) to optimize the model within a simulated O&M environment, thereby reducing hallucinations by directly rewarding correct outcomes. R-Log is first cold-started on a curated dataset of 2k+ reasoning trajectories, guided by 13 strategies from manual O&M practices, to establish an initial reasoning capability. This ability is then refined via RL using a joint reward function. Empirical evaluations on real-world logs show that R-Log outperforms existing methods across five log analysis tasks, particularly in unseen scenarios (by 228.05%). We also designed R-Log-fast with 5x speedup while keeping 93% of the efficacy.