For the cake: Place an oven rack in the center of the oven. Preheat the oven to 350 degrees F. Butter and flour a 9 by 5-inch loaf pan. Set aside.
In a medium bowl, combine the flour, baking powder, salt, cocoa powder, cinnamon, ginger, nutmeg, and allspice.
In a large bowl, beat the sugars, oil, applesauce, eggs, and vanilla extract to blend. Add the dry ingredients and stir until just blended. Pour the batter into the prepared pan and smooth the top with a spatula. Bake for 40 to 45 minutes or until a toothpick inserted into the center of the cake comes out clean. Cool the cake for 10 minutes. Unmold the cake and place on a wire rack to cool completely. Dust with powdered sugar and cut into slices for serving.