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I am working on problems such as :

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The charges on the side groups at each pH are found by the following rule:

  • At $\ce{pH < pK}$ a , ${H+}$ on, protonated
  • At $\ce{pH > pK}$, ${H+}$ off, deprotonated
  • at $\ce{pH = pKa}$ , neutral

But why can we compare $\ce{pH}$ and $\ce{pKa}$ directly?

I just started my amino acids topic. In all my acid and base questions we always compared $\ce{pH}$ to $\ce{pKa}$ using the Henderson Hasselbalch Equation. So why do we now compare $\ce{pH}$ to $\ce{pKa}$ directly?

Henderson Hasselbalch Equation

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The equation you wrote shows a close relationship between pKa aand pH: when $\ce{log ([A^{-}]/[HA])=0}$, pKa=pH.

$\ce{log ([A^{-}]/[HA])}$ equals zero when $\ce{[A^{-}]/[HA]}$ equals 1: this means that at the equivalence point, pH equals pKa.

The properties you listed are correct, and all of them come from this relationship.

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