Large Language Models (LLMs) have become a focal point of research across various domains, including software engineering, where their capabilities are increasingly leveraged. Recent studies have explored the integration of LLMs into software development tools and frameworks, revealing their potential to enhance performance in text and code-related tasks. Log level is a key part of a logging statement that allows software developers control the information recorded during system runtime. Given that log messages often mix natural language with code-like variables, LLMs' language translation abilities could be applied to determine the suitable verbosity level for logging statements. In this paper, we undertake a detailed empirical analysis to investigate the impact of characteristics and learning paradigms on the performance of 12 open-source LLMs in log level suggestion. We opted for open-source models because they enable us to utilize in-house code while effectively protecting sensitive information and maintaining data security. We examine several prompting strategies, including Zero-shot, Few-shot, and fine-tuning techniques, across different LLMs to identify the most effective combinations for accurate log level suggestions. Our research is supported by experiments conducted on 9 large-scale Java systems. The results indicate that although smaller LLMs can perform effectively with appropriate instruction and suitable techniques, there is still considerable potential for improvement in their ability to suggest log levels.