Preheat the oven to 350°F (175°C). Grease and flour a 9×13 inch baking pan.
In a medium bowl, whisk together the all-purpose flour, baking powder, baking soda, and salt. Set aside.
In a large bowl, cream the softened unsalted butter and granulated sugar together until light and fluffy, about 3-4 minutes using an electric mixer.
Beat in the eggs one at a time, mixing well after each addition. Stir in the vanilla extract.
Add the mashed ripe bananas to the butter mixture and mix until combined.
Alternately add the dry flour mixture and buttermilk to the banana mixture, beginning and ending with the flour mixture. Mix until just combined; do not overmix.
Pour the batter evenly into the prepared baking pan and smooth the top with a spatula.
Bake in the preheated oven for 40-45 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.