This paper introduces Logarithmic Posits (LP), a novel data type that combines the adaptability of posits with the hardware efficiency of logarithmic number systems (LNS). LP dynamically adjusts to DNN weight and activation distributions by parameterizing bit fields, offering a wider dynamic range and higher accuracy compared to traditional quantization methods. The authors develop LP Quantization (LPQ), a genetic algorithm-based framework that optimizes layer-wise LP parameters while reducing representational divergence between quantized and full-precision models. Additionally, they propose a unified mixed-precision LP accelerator (LPA) architecture, which integrates LP processing elements (PEs) into a systolic array. Extensive experiments on various CNNs and ViT models demonstrate that the co-design achieves an average top-1 accuracy drop of <1%, 2× improvement in performance per unit area, and 2.2× gain in energy efficiency compared to state-of-the-art quantization accelerators. The code for LP is available at <https://github.com/georgia-tech-synergy-lab/LogarithmicPosit>.This paper introduces Logarithmic Posits (LP), a novel data type that combines the adaptability of posits with the hardware efficiency of logarithmic number systems (LNS). LP dynamically adjusts to DNN weight and activation distributions by parameterizing bit fields, offering a wider dynamic range and higher accuracy compared to traditional quantization methods. The authors develop LP Quantization (LPQ), a genetic algorithm-based framework that optimizes layer-wise LP parameters while reducing representational divergence between quantized and full-precision models. Additionally, they propose a unified mixed-precision LP accelerator (LPA) architecture, which integrates LP processing elements (PEs) into a systolic array. Extensive experiments on various CNNs and ViT models demonstrate that the co-design achieves an average top-1 accuracy drop of <1%, 2× improvement in performance per unit area, and 2.2× gain in energy efficiency compared to state-of-the-art quantization accelerators. The code for LP is available at <https://github.com/georgia-tech-synergy-lab/LogarithmicPosit>.