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 mashed bananas to the mixture and mix well.
In another bowl, sift together the flour, baking soda, and salt.
Gradually add the dry ingredients to the banana mixture, mixing just until combined.
Stir in the vanilla extract.
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.