Flexible DC transmission systems based on modular multilevel converters (MMC) typically use traditional sliding mode observation techniques to observe state variables, which can significantly reduce the accuracy of system state reconstruction and transient response speed under sudden changes in operating conditions. A discrete-time logarithmic sliding mode observation technique (DTLSMO) is proposed, which constructs dynamic boundary layer functions of observation error norm and expected amplitude, achieves adaptive nonlinear mapping of observer gain, and avoids the parameter tuning process that relies on empirical trial and error. Establish a discretization model of MMC under parameter mismatch and analyze its observability. Design a logarithmic sliding mode approaching law based on active damping characteristics and inject it into the bridge arm dynamic equation. Use the second harmonic circulating current injection algorithm to achieve the coordinated optimization of capacitor voltage balance control and circulating current suppression. Prove the asymptotic stability and generalized convergence of the proposed observer. Analyze the DC voltage 9.6 kV MMC simulation system on MATLAB/Simulink platform and build a hardware prototype for experimental verification. Introduce evaluation indicators such as ITAE to compare and analyze the dynamic performance differences between DTLSMO and traditional time discrete sliding mode observer (DTSMO) and adaptive observer (AO). It is proved that the proposed observer improves the observation accuracy by 0.935% and 2.535% respectively when the bridge arm inductance is mismatched by 20%. It can still demonstrate its fast response ability under uncertain capacitor values and sudden changes in DC voltage, and has good robustness.