In this paper, we account for the many critical exponents derived from the studies of the electrical conductivity in porous media by applying analysis of the well-known relation known as Archie's law. In spite of its seeming simplicity this law is considered to be "poorly understood," and the question that was and still is debated in the literature is whether there is some "hidden physics" in this law, or if it is "strictly a parametrization use for curve fitting with a priori no physical meaning." Our solution to the corresponding long-debated 78 years old puzzle is based on the classical percolation theory, but it also involves a principle that is based on continuum percolation. This principle is that the electrical properties of a percolation system are determined by the interplay between the connectivity of the conducting objects in that system, and the connectivity of the intersections between pairs of them. We thus propose a general concept that we call an electrically affected connectivity, and we predict the corresponding evolvement of the conductivity critical exponent with the increase of the content of the electrically conducting phase. Then, we show that the zerolike threshold that characterizes Archie's law is what enables the observation of this evolution. Combining the above principle and the latter feature, we provide a holistic, yet simple, solution to the longstanding controversy surrounding this law and its practical applications. In contrast with many previous claims that Archie's law lacks a physical basis, and the commonly suggested experiential explanations for it, we provide a solution that is physically based and thus elucidates Archie's law by showing clearly that it represents a bona fide phase transition phenomenon. This conclusion and its generality are strongly supported by the fact that it also explains the behavior of the electrical conductivity exponents in nonporous systems such as composite materials. The predicted ability to extract the long sought microgeometrical information from Archie's-law data, within the framework of the percolation phase transition, is expected to open a new direction in the understanding and the applications of this law.