Empirical power law expressions of the form NCCN = Csk have been used in cloud physics for over 4 decades to relate the number of cloud condensation nuclei (CCN) and droplets formed on them to cloud supersaturation, s. The deficiencies of this parameterization are that the parameters C and k are usually constants taken from the empirical data and not directly related to the CCN microphysical properties and this parameterization predicts unbounded droplet concentrations at high s. The activation power law was derived in several works from the power law Junge-type aerosol size spectra and parameters C and k were related to the indices of the power law, but this still does not allow to describe observed decrease of the k-indices with s and limited NCCN. Recently, new parameterizations for cloud drop activation have been developed based upon the lognormal aerosol size spectra that yield finite NCCN at large s, but does not explain the activation power law, which is still traditionally used in the interpretation of CCN observations, in many cloud models and some large-scale models. Thus the relation between the lognormal and power law parameterizations is unclear, and it is desirable to establish a bridge between them. In this paper, algebraic and power law equivalents are found for the lognormal size spectra of partially soluble dry and wet interstitial aerosol, and for the differential and integral CCN activity spectra. This allows derivation of the power law expression for cloud drop activation from basic thermodynamic principles (Köhler theory). In the new power law formulation, the index k and coefficient C are obtained as continuous analytical functions of s and expressed directly via parameters of aerosol lognormal size spectra (modal radii, dispersions) and physicochemical properties. This approach allows reconciliation of this modified power law and the lognormal parameterizations and their equivalence is shown by the quantitative comparison of these models as applied to several examples. The advantages of this new power law relationship include bounded Nd at high s, quantitative explanation of the experimental data on the k-index and possibility to express k(s) and C(s) directly via the aerosol microphysics. The modified power law provides a framework for using the wealth of data on the C, k parameters accumulated over the past decades, both in the framework of the power law and the lognormal parameterizations.