Preheat oven to 350 degrees F (175 degrees C). Grease and flour a 9x9-inch pan.
Beat butter and sugar with an electric mixer in a large bowl until light and fluffy. The mixture should be noticeably lighter in color. Add room-temperature eggs one at a time, allowing each egg to blend into butter mixture before adding the next. Beat in vanilla extract with last egg. Combine flour and baking powder, and add to butter mixture, mixing well. Stir in milk until batter is smooth. Pour or spoon batter into prepared pan.
Bake in preheated oven until cake springs back to the touch, 30 to 40.