In the present paper, we study some probabilistic and statistical properties of the Markov-switching asymmetric log GARCH processes, where the log −volatility follows a standard asymmetric log GARCH process for each regime. In these models, the coefficients of log −volatility depend on the state of a non-observed Markov chain. The main motivations of this new model can capture the asymmetries and hence leverage effect. Additionally, The volatility coefficients are not subject to positivity constraints. Therefore, some probabilistic properties of Markov-switching asymmetric log GARCH models have been obtained, especially, sufficient conditions ensuring the existence of stationary, causal, ergodic solution and moments properties are given. Furthermore, we show the strong consistency of the quasi-maximum likelihood estimator (QMLE) under mild assumptions. Finally, we provide a simulation study of the performance of the proposed estimation method and the MS − log GARCH is applied to model the exchange rate of the Algerian Dinar against the US-dollar.