Step 1: Why we need the Nernst equation. The standard potential \(E^{\circ}_{cell}\) applies only when every ion is at unit concentration (1 M). Real cells rarely satisfy this, so Nernst gave a correction that folds in the actual concentrations through the reaction quotient \(Q\).
Step 2: The equation. \( E_{cell} = E^{\circ}_{cell} - \dfrac{2.303RT}{nF}\log Q \), which at 298 K becomes \( E_{cell} = E^{\circ}_{cell} - \dfrac{0.0591}{n}\log Q \).
Step 3: Approaching equilibrium. As the reaction runs, reactants fall and products rise, so \(Q\) climbs and \(E_{cell}\) steadily drops. The instant equilibrium is reached the driving force vanishes, hence \(E_{cell}=0\) and \(Q\) equals the thermodynamic equilibrium constant \(K_c\).
Step 4: Deriving the link. Setting \(E_{cell}=0\) and \(Q=K_c\) in the Nernst equation and solving for the standard potential gives \( \log K_c = \dfrac{nE^{\circ}_{cell}}{0.0591} \), so a bigger cell EMF corresponds to a much larger equilibrium constant.
\[\boxed{\log K_c = \frac{nE^{\circ}_{cell}}{0.0591}}\]