From all the data I have worked with, I have noticed that in a linear model with one explanatory variable, taking the ln of that explanatory variable and using the result as the new "independent" variable in the model makes the regression line (with the estimators for the parameters gotten through OLS) a better fit (in terms of a better R^2 value).
Assuming the X's being considered are always positive, my question just has three parts:
1) Is it always the case that the new regression line for "Y on ln(X)" has a more favorable R^2 value as compared to the old regression line for "Y on X"?
2) If no, when is it not?
3) What determines this change in the R^2 value exactly?
All help would be appreciated!
(I know I mentioned ln(X) in this particular question but naturally this doubt would be applicable to most logs with any base, barring some cases)