This paper presents a framework based on the logarithmic number system to implement adaptive filters with error nonlinearities in hardware. The framework is demonstrated through pipelined implementations of two recently proposed adaptive filtering algorithms based on logarithmic cost, namely, least mean logarithmic square (LMLS) and least logarithmic absolute difference (LLAD). To the best of our knowledge, the proposed architectures are the first attempts to implement both LMLS and LLAD algorithms in hardware. We derive error computing algorithms to realize the nonlinear error functions for LMLS and LLAD and map them onto hardware. We also propose a novel variable-α scheme to enhance the original LMLS algorithm and prove its robustness and suitability for VLSI implementations in practical applications. Detailed bit width and error analysis are carried out for the proposed VLSI fixed point implementations. Postlayout implementation results show that with an additional multiplier over conventional least mean square (LMS), 7-dB improvement in steady-state mean square deviation performance can be achieved and with the proposed variable-α scheme, 12-dB improvement can be achieved without compromising the convergence. We will show that LMLS can potentially replace LMS in practical applications, by demonstrating a proof-of-concept by extending the framework to transform domain adaptive filters. © 2017 IEEE.