How to get a dataframe values in one single column for the following dataset?
You are not getting 2 columns. You are getting 2 rows.

The function alpha = (np.log([dataset.m38,dataset.m78])/np.math.log(38,78.7)) returns a 2-dimensional array. Putting in:

         m38     m78
      0  4.4717  4.8745
      1  4.4569  4.6491
      2  4.5101  4.7262
      3  4.4407  4.8234
      4  4.1184  4.3862


gives us: [[0.41174795 0.41083658 0.4140986  0.40983552 0.38912198]
           [0.43545842 0.42244323 0.42696487 0.43256132 0.40644075]]

This is due to the fact that np.log([dataset.m38,dataset.m78]) returns a 2-dimensional array.

Regarding the flipped columns/ row issue: Once your function doesn't return a 2-dimensional array, it will most likely return a Pandas Series which will have one column and 8000 rows.

If you want to turn that into a DataFrame, you can use 
alpha = pd.DataFrame({'alpha':alpha})

Think about what your formula does. Putting in [dataset.m38,dataset.m78] will do the calculation for both columns seperately resulting in 2 different arrays.