Preheat oven to 350°F. Butter and flour a Bundt pan.
Combine flour, baking soda, baking powder, cinnamon, nutmeg and salt in a medium bowl. Set aside.
Using an electric mixer, beat together, sugar and butter until fluffy, about 2 minutes. Stir in eggs one at a time. Mix thoroughly to combine. Add flour mixture; beat just until blended. Stir in chopped apples. Transfer batter to prepared pan.
Bake cake until toothpick inserted into center comes out clean, about 45-55 minutes. Cool cake in pan on rack.