Preheat your oven to 350°F (175°C). Grease and flour a baking pan.
In a mixing bowl, whisk together the flour, baking soda, cinnamon, nutmeg, and salt.
In another bowl, combine the applesauce, sugar, brown sugar, oil, and eggs. Mix well.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined.
Pour the batter into the prepared baking pan and spread evenly.
Bake for 45 minutes or until a toothpick inserted into the center comes out clean.
Allow the cake to cool in the pan for 10 minutes, then transfer to a wire rack to cool completely.