The effects of higher-order contributions to the linearized renormalization group equations in critical phenomena are discussed. This analysis leads to three quite different results: (i) An exact scaling law for redefined fields is obtained. These redefined fields are normally analytic functions of the physical fields. Corrections to the standard power laws are derived from this scaling law. (ii) The theory explains why logarithmic terms can exist in the free energy. (iii) The case in which the energy scales like the dimensionality is analyzed to show that quite anomalous results may be obtained in this special situation.