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The conjugate base of a strong acid is stable as is, so it does not result in the deprotonization of water which would result in a change in pH. However, the conjugate base of a weak acid is unstable and will therefore pull an H+ from H2O to form an acid and OH- in solution, thereby increasing the pH.
A strong acid will have a weak conjugate base and a strong base will have a weak conjugate acid. Weak acids tend to have stronger conjugate bases, and weak bases tend to have stronger conjugate acids. What this basically means is that with a strong acid, its conjugate base is unlikely to be a good proton acceptor, but with a weak acid, its conjugate base is much more likely to accept a proton (which is why the majority of the weak acid molecules remain in their original form). This same concept can be applied to bases.
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