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The partition coefficient is a very useful metric in characterizing compounds particularly drugs. What chemical properties influence that partitioning? Are chemical moieties additive in their LogP? What if the compound has both hydrophobic and hydrophilic ends?

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The Wikipedia page you linked to has some interesting information, that could at least get you started:

It has been shown that the log P of a compound can be determined by the sum of its non-overlapping molecular fragments (defined as one or more atoms covalently bound to each other within the molecule). Fragmentary log P values have been determined in a statistical method analogous to the atomic methods (least squares fitting to a training set). In addition, Hammett type corrections are included to account of electronic and steric effects. This method in general gives better results than atomic based methods, but cannot be used to predict partition coefficients for molecules containing unusual functional groups for which the method has not yet been parameterized (most likely because of the lack of experimental data for molecules containing such functional groups).

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Partition Coefficient, represented as P and defined as: P = [Organic phase] / [Aqueous phase]

Where "[]" represents concentration in the phase and $\mathrm{\log (P)= \log_{10}(Partition Coefficient)}$ if log P value is higher then its more lipophilic ;those drugs were more absorbed by cells and elimination of those drugs from body will be less .The partition coefficient is a ratio of concentrations of un-ionized compound between the two solutions so for a chemical which is having both hydrophobic and hydrophilic ends also have log P values.

Eg: Oleic acid: $\mathrm{\log (P)= 7.64}$

chemicals which are hydrophopic(lipophilic) tends to have more Log P values and hydrophilic(lipophobic) chemicals have less log P.

eg: pentane($\mathrm{\log (P)= 3.45}$) [lipohilic], 1-pentanol ($\mathrm{\log (P)= 1.51}$) [lipophobic]

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