Log scaling whole dataset
I would recommend taking a look at the transformation you did:

Consider whether you would want to add a really small number like 0.0000001 to zero before taking the log. Probably not, as you will get a large negative number. However, there should be little to no difference in results once you start adding large enough numbers like 0.1. I would recommend trying each and looking at a density plot of the resulting data. Investigate which distribution looks the least skewed and go with it.