Heat oven to 325°F. Use butter to grease a loaf pan and set it aside.
In a mixing bowl, sift together the flour, baking soda, baking powder, and salt. Set aside.
In a large mixing bowl, add the butter and brown and granulated sugars. Beat until creamed and a bit fluffy.
Add the beaten eggs, mashed bananas, and vanilla into the butter/sugar mixture. Mix well.
Add the dry ingredients to the wet ingredients. Gently stir by hand to the point that the dry ingredients are incorporated, but JUST moistened. It's okay to have a few small streaks of flour showing. Don't over mix!
Pour the batter into the prepared loaf pan. Pleace in the preheated oven and bake for 60-70 minutes, or until a toothpick or knife inserted into the center of the loaf comes out clean.
Remove from the oven and set the pan on a rack to cool.
Remove the banana bread loaf from the pan when cooled or mostly cool. You can slice it and serve warm with butter spread on the slices, or let cool completely and wrap in plastic wrap to store.