Isaiah 10:15 (a) This is a type of person or a nation whom GOD uses to punish other people or nations. Assyria was the ax in this case ( Isaiah 10:5) and GOD's hand and arm wielded the ax. Assyria is also compared to a saw, a rod, and a staff. All of these are just different aspects of the one nation of Assyria in the hand of GOD for the punishment of others. (See under "BATTLE AX").
Matthew 3:10 (b) Here the ax is a type of GOD's judgment which will cut down the nation of Israel because of their disobedience and fruitlessness. This ax of GOD's wrath also cuts down individuals whose lives are not profitable to GOD. (See also Jeremiah 51:20).