In this work we apply Thompson's scaling approach (of dimensions) to study the scalar field theories Φn. This method can be considered as a simple and alternative way to the renormalization group (RG) approach and when applied to the Φn Lagrangian is able to obtain the coupling constant behavior g(μ), namely the dependence of g on the energy scale μ. The calculations are evaluated just at [Formula: see text], where the dimension dc is similar to a kind of upper critical dimension of the problem, or in other words the dimension where the Φn theory becomes renormalizable, so that we obtain logarithmic behavior of the coupling g at dc. Due to the universal logharithmic behavior of the coupling g at dc for any value of n in the Φn theory, we are able to estimate a certain βn function given in a closed form, which is a novelty obtained by the present method.