I really want to know why in the result of SDS PAGE, log of molecular weight(MW) and migration distance (distance from the loading well) have a linear relationship. Why is it log(MW) instead of MW? Thank you!
An SDS-PAGE standard curve of Rf vs LogMW is linear because the standard you're given is engineered to behave that way. An expanded SDS-PAGE curve would show that the actual graph is sigmoidal: Structues that are too large can't enter the matrix, and too small they pass right through. Your standard has taken into account the % composing the gel, and the size of the proteins which the standard consists of that ensure you get the linear data points on the semilog graph.
If we look at the raw data though plotted linearly, the graph isn't entirely linear. The data is much prettier, and trivial to convert back to MW if we log transform the MW. It's the same data, just mathematically transformed. Some more reading on it here.