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Do not confuse acid and base strength with concentration. Increasing or decreasing the concentration of an acid or base does not change its strength. Don't think of acid or base solutions like you might think of a cup of coffee where the more beans you use the stronger the coffee will be. The acid or base strength is an intrinsic property of the molecule and depends on how much the molecule reacts with water to either transfer or accept a proton.
Remember this approximation: In most weak acid or weak base solutions the initial concentration of acid or base doesn't change significantly. However, some amount of the acid or base must dissociation to produce significant concentrations of the resulting conjugate partner and H3O+ or OH-. Confused?
This apparent paradox can be reconciled if you remember that, like Einstein said, it's all relative. The initial concentration of a weak acid or weak base does not change significantly when Ka or Kb is small relative to the initial concentration of weak acid or weak base. The concentration of the conjugate partner produces do change significant relative to their starting concentrations which is zero. Everything is significant compared to zero!
The idea that salts can be acids or bases is a bit strange. Can't we just name salts salts and leave it at that? After all, the salts themselves are usually just boring white powders or granules. Keep in mind: it isn't really the salt itself that is basic or acidic. It is the cations and anions released after the salt dissolves that are the real acids and bases. For many salts, like NaCl, the cations and anions don't mess with protons of the solution so the salt doesn't change the solution pH. Some salts, however, do release cations and anions that fight over the protons, sometimes resulting in major changes to the solution acidity or basicity.