Preheat your oven to 350°F (180°C). Grease and flour a 9-inch (23 cm) cake pan. In a medium bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, and salt.
In a large bowl, cream the butter and sugar until light and fluffy. Beat in the eggs one at a time, then stir in the applesauce and vanilla extract.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Pour the batter into the prepared pan and smooth the top with a spatula.
Bake for 35-40 minutes, or until a toothpick inserted in the center of the cake comes out clean. Let the cake cool in the pan for 10 minutes, then transfer it to a wire rack to cool completely. Enjoy your delicious applesauce cake!