In this work, we study the inconsistency of EKF-based SLAM from the perspective of observability. We analytically prove that when the Jacobians of the system and measurement models are evaluated at the latest state estimates during every time step, the linearized error-state system employed in the EKF has observable subspace of dimension higher than that of the actual, nonlinear, SLAM system. As a result, the covariance estimates of the EKF undergo reduction in directions of the state space where no information is available, which is a primary cause of the inconsistency. Furthermore, a new "First-Estimates Jacobian" (FEJ) EKF is proposed to improve the estimator's consistency during SLAM. The proposed algorithm performs better in terms of consistency, because when the filter Jacobians are calculated using the first-ever available estimates for each state variable, the error-state system model has an observable subspace of the same dimension as the underlying nonlinear SLAM system. The theoretical analysis is validated through both simulations and experiments.