A chemist needs an acetic acid buffer of pH 5.20…

A buffer is a weak acid with its salt, or a weak base with its salt. $\text{pH} = \text{p}K_a + \log\dfrac{[\text{salt}]}{[\text{acid}]}$ and $\text{pOH} = \text{p}K_b + \log\dfrac{[\text{salt}]}{[\text{base}]}$. Added $\text{H}^+$ turns the salt's anion into the weak acid; added $\text{OH}^-$ turns the weak acid into its anion. A common ion pushes back the ionisation of a weak acid.