Hi, LogP is a partition coefficient between octanol and water. You need to have 2 phases, one organic and water for the other. Thus, solvent miscible in water, like methanol, are not usable for this purpose.
I agree with Gilles's explanation. Moreover, octanol is the standard organic phase for log P calculations. All standard logP values are given in literature has been calculated with respect to the distribution of the substances in octanol/water binary mixture.
As Gilles said, you cant use organic solvents that are miscible with water because then you cannot see the “partioning”.
Octanol is the chosen standard because: a) it is immiscible in water and b) it mimics the phospholipids of biological membranes. Thus the partioning of a compound between water and octanol can help predict a compounds fate in the environment or a drugs fate in the body.