Preheat the oven to 350°F (175°C). Grease and flour a 9-inch round cake pan.
In a medium bowl, whisk together the all-purpose flour, baking soda, and salt. Set aside.
In a large bowl, combine the granulated sugar, brown sugar, and vegetable oil. Mix until well combined.
Add the eggs one at a time to the sugar and oil mixture, beating well after each addition.
Stir in the vanilla extract.
Add the mashed bananas, plain yogurt, and milk to the wet ingredients and mix until combined.
Gradually add the dry ingredients to the wet ingredients, stirring gently until just combined. Do not overmix.
Pour the batter into the prepared cake pan and smooth the top with a spatula.
Bake in the preheated oven for 35 to 40 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and allow it to cool in the pan for 10 minutes.
Transfer the cake to a wire rack to cool completely before serving.