The Logistic Mapping is studied from a particular perspective. Firstly an identification is made of this mapping with the Newton iterating mapping to find the roots of a corresponding function. With this approach one identifies the fixed points and the periodic points of a stable cycle as zeros of that function, and the unstable fixed points and periodic cycles as singularities of the same function. The bifurcation behavior is provided by the sign of the exponents that turn from stable to unstable, the bifurcation appears as any of them becomes infinite.The periodic cycles of periods one, two, three and four are explicitly considered, determining the sets of stability and the values when the new stables cycles arise, simultaneously becoming unstable the previous periodic points.