Preheat oven to 350 degrees F (175 degrees C). Grease and flour a 9x13 inch pan.
In a medium bowl, combine the flour, baking powder, baking soda and salt. In a large bowl, cream together the butter and sugar until light and fluffy. Beat in the eggs one at a time, then stir in the vanilla. Gradually beat in the dry ingredients until just blended. Finally stir in the applesauce.
Pour batter into prepared pan. Bake for 35 to 40 minutes in preheated oven, or until a toothpick inserted into center of cake comes out clean.