Preheat the oven to 350°F (175°C). Grease and flour a 9-inch round cake pan.
In a large mixing bowl, whisk together the all-purpose flour, baking powder, baking soda, salt, ground cinnamon, and ground nutmeg.
Add the granulated sugar and light brown sugar to the dry ingredients and whisk to combine.
Add the eggs, vegetable oil, Greek yogurt, and vanilla extract to the bowl. Stir with a wooden spoon or spatula until just combined and a smooth batter forms.
Fold the diced apples gently into the batter until evenly distributed.
Pour the batter into the prepared cake pan and spread it out evenly.
Bake in the preheated oven for 40 to 45 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and let it cool in the pan for 10 minutes.
Transfer the cake to a wire rack to cool completely before slicing and serving.