The demand for explainable and energy-efficient artificial intelligence (AI) systems for edge computing has led to growing interest in electronic systems dedicated to Bayesian inference. Traditional designs of such systems often rely on stochastic computing, which offers high energy efficiency but suffers from latency issues and struggles with low-probability values. Here, we introduce the logarithmic memristor-based Bayesian machine, an innovative design that leverages the unique properties of memristors and logarithmic computing as an alternative to stochastic computing. We present a prototype machine fabricated in a hybrid CMOS/hafnium-oxide memristor process. We validate the versatility and robustness of our system through experimental validation and extensive simulations in two distinct applications: gesture recognition and sleep stage classification. The logarithmic approach simplifies the computational model by converting multiplications into additions and enhances the handling of low-probability events, which are crucial in time-dependent tasks. Our results demonstrate that the logarithmic Bayesian machine achieves superior performance in terms of accuracy and energy efficiency compared to its stochastic counterpart, particularly in scenarios involving complex probabilistic models. This approach enables the development of energy-efficient and reliable AI systems for edge devices.