By definition soap has to be made with lye (Caustic Soda + Water). Soap made without Lye is a Syndet Bar (Synthetic Soap).
Now, just because a soap starts with Fatty Acids (vegetable Oils) + Caustic Soda (Lye) does NOT MEAN THERE IS LYE IN THE SOAP. In fact, assuming the correct recipes are followed there can be NO caustic left because of the change in the oils into Sodium Salts. This is sometimes shown on soap labels as Saponified Vegetable Oils.
Real soap made the natural way must start with a strong alkali (sodium hydroxide for solid soap and Potassium Hydroxide if you want to make liquid soap).
The myth by people who sell Synthetic Soap or Surfactant based liquid soap is that natural soap has lye in it. In truth that is like saying you will find egg yokes in cake made with eggs. The cake mixing and cooking process changes the ingredients into something nice to eat. If you make the cake properly you will not finds eggs or flour.
I hope this helps
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