Preheat the oven to 350°F (175°C). Grease a 9×5-inch loaf pan or line it with parchment paper.
In a medium bowl, whisk together the all-purpose flour, baking powder, baking soda, salt, and ground cinnamon until well combined.
In a large bowl, whisk the granulated sugar, light brown sugar, unsweetened applesauce, vegetable oil, eggs, and vanilla extract until smooth and fully combined.
Gradually add the dry ingredients to the wet ingredients, stirring gently just until combined. Do not overmix.
Fold in the diced apple pieces evenly into the batter.
Pour the batter into the prepared loaf pan and spread it out evenly.
Bake in the preheated oven for 45 to 50 minutes, or until a toothpick inserted into the center comes out clean.
Remove the loaf from the oven and allow it to cool in the pan for 10 minutes.
Transfer the loaf to a wire rack and cool completely before slicing.
Slice into 8 even pieces and serve.