Derive Henderson – Hasselbalch equation?

We can derive a general equation for calculating the pH of any buffer solution. Consider a mixture of weak acid (HX) and its conjugate base.
HX + H₂O ⇌ H₃O⁺ + X^–
K_a = [H₃O⁺] [X^–] / [HX]
Taking log on both the sides, we will get:
Log K_a = log [H₃O⁺] + log ([X^–] / [HX])
-log K_a = – log [H₃O⁺] – log ([X^–] / [HX])
pK_a = pH – log ([X^–] / [HX])
pH = pK_a + log ([X^–] / [HX])

The quantity [X^–] / [HX] is the ratio of concentration of salt or conjugate base of acid and acid present in the mixture.
Therefore, pH = pK_a + log ([salt]/ [base])
This equation is called Henderson – Hasselbalch equation. Similarly, for a basic buffer,
pOH = pK_b + log ([salt]/ [base])
pH = 14 – pK_b – log ([salt]/ [base])