Preheat the oven to 350°F (175°C). Grease and flour a 9×5-inch loaf pan.
In a large mixing bowl, combine the mashed bananas, granulated sugar, brown sugar, and melted butter. Mix well until smooth.
Add the eggs one at a time to the banana mixture, beating well after each addition.
Stir in the vanilla extract.
In a separate bowl, whisk together the all-purpose flour, baking soda, baking powder, and salt.
Gradually add the dry ingredients to the wet banana mixture, stirring gently until just combined.
Fold in the Greek yogurt until the batter is smooth and well incorporated.
Pour the batter into the prepared loaf pan and spread 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 banana cake loaf to a wire rack to cool completely before slicing.