Question regarding chemical equilibrium when adding a salt to acid solution
the second relation is wrong to : $pK_{a} = -\log\frac{[\ce{H+}]_{eq}[\ce{A-}]_{eq}}{[\ce{HA}]_{eq}}=-\log[\ce{H+}]_{eq} -\log\frac{[\ce{A-}]_{eq}}{[\ce{HA}]_{eq}}$ so $pH=pK_a+\log\frac{[\ce{A-}]_{eq}}{[\ce{HA}]_{eq}}$
the acidity constant is the relation which links the concentrations of the species present (and not the reverse): if we add the salt $ \ce {A^-} $, then according to the relation you see that the pH increases. You can also see it by reasoning thus: you add a base so the pH can only increase.