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A base is any substance that neutralizes acid. It may or may not be soluble in water. 

Those bases which are soluble in water are called alkalis. 

Thus an alkali is a water-soluble base. For example NaOH, KOH etc.

Alkali, as well as base neutralize acid, but we have observed that alkali produce OH ions in the water, whereas bases such as copper oxide and ferrous do not produce OH ions.

This is why it is said that all alkalis are bases but all bases are not alkalis.

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S.No Acids Bases 
 
1. Its taste is sour. Its taste is bitter.
2. When added to litmus solution, it turns red. When added to litmus solution, it turns blue.
3. Examples: Curd, lemon juice and vinegar Examples: Baking soda, Soap and lime water.

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