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 to 4 minutes.
Add the eggs one at a time, beating well after each addition. Stir in the vanilla extract.
Mix in the mashed ripe bananas until combined.
Alternately add the flour mixture and the buttermilk to the banana mixture, beginning and ending with the flour mixture. Mix just until combined; do not overmix.
Pour the batter into the prepared baking pan and spread evenly.
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.
Turn the cake out onto a wire rack and cool completely before serving or frosting.