The 1/2 equivalent point in a titration curve

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At the half-equivalence point of a titration, the concentrations of the remaining non-neutralized titrated acid and its conjugate base are equal. This results in a concentration ratio of 1. According to the Henderson-Hasselbalch equation, the logarithm of this ratio (log(1)) is 0, leading to the conclusion that at this point, the pH equals the pKa of the acid.
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It is pretty challenging for me to understand how the 1/2 equivalent point in a titration curve (when either acid or base is weak) the pH = pKa.
I researched about the formula that is used to get this, but I couldn't quite understand that.
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What are expected concentrations of the remaining, non-neutralized titrated acid and its conjugate base at half point?

What is ratio of these concentrations equal to?

What is log of that?
 
From the equation above (the Henderson-Hasselbach equation if I remember correctly), the half equivalent point is when the concentrations of the acid and its conjugate base are equal. Any number divided by itself is 1 (one). The log of 1 is 0 (zero), so therefore pH = pKa
 

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