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A detailed analysis of the electric field integral equation (EFIE) at low frequencies is presented. The analysis, that is based on the Sobolev space mapping properties of the EFIE, explains the conditioning growth of the majority of currently available quasi-Helmholtz decomposition methods (such as Loop-Tree/Star, Loop-Rearranged Trees, etc.) that cure the EFIE low-frequency breakdown. It is shown that these methods have a conditioning that grows polynomially with the number of unknowns. To solve this problem, in this work we present a quasi-Helmholtz decomposition method that leads to an EFIE whose conditioning grows only logaritmically with the number of unknowns. This result is obtained by properly regularizing both the solenoidal and non-solenoidal part of the EFIE. The regularization is obtained by introducing a new set of loop hierarchical basis functions. Numerical tests are provided to confirm the results obtained by the theory.