What is the P in LogP?
It means the following:

$$\log P = \log(P) = \log_{10}(P) = \log_{10} \left(\frac{C_{\mathrm{octanol}}}{C_{\mathrm{water}}} \right) $$

With $C_{\mathrm{octanol}}$ being the concentration of the solute in 1-octanol and $C_{\mathrm{water}}$ the concentration of the solute in water.