Preheat your oven to 350°F (175°C).
In a mixing bowl, cream together the butter and sugar until light and fluffy.
Add the beaten eggs and vanilla extract, and mix well.
Stir in the mashed bananas.
In another bowl, whisk together the flour, baking soda, and salt.
Gradually add the dry ingredients to the banana mixture, mixing until just combined.
Pour the batter into a greased baking pan.
Bake for 45 minutes, or until a toothpick inserted into the center comes out clean.
Let the cake cool in the pan for 10 minutes, then transfer to a wire rack to cool completely.