If revenge is sin, then how can God take vengeance?

Jeremiah 51:6, 56Flee from Babylon! Save yourselves! Don’t get trapped in her punishment! It is the Lord’s time for vengeance; he will repay her in full. . . . Destroying armies come against Babylon. Her mighty men are captured, and their weapons break in their hands. For the Lord is a God who gives just punishment; he always repays in full.

Jonah 3:10When God saw what they had done and how they had put a stop to their evil ways, he changed his mind and did not carry out the destruction he had threatened.

Nahum 1:2-8The Lord is a jealous God, filled with vengeance and rage. He takes revenge on all who oppose him and continues to rage against his enemies! The Lord is slow to get angry, but his power is great, and he never lets the guilty go unpunished. . . . The Lord is good, a strong refuge when trouble comes. He is close to those who trust in him. But he will sweep away his enemies in an overwhelming flood.

Can a loving, merciful, and compassionate God also be vengeful? The answer lies in a proper understanding of God’s character. Because God is holy, he cannot tolerate sin and evil and therefore must punish all unrighteousness. Because he is merciful, he has not yet destroyed the wicked but allows them the opportunity for repentance and forgiveness. From a human viewpoint, God might appear vengeful. From the viewpoint of true holiness, God is merely administering justice, as is his right.