Preheat the oven to 350F.
Prepare the bundt pan by coating the inside with butter and then dusting with flour. Ensure that every corner of the pan is well covered to prevent the cake from sticking later.
Zest 3 of the lemons and set aside the zest. Then, remove the white pith from each lemon. Place the fruit in a food processor and blend until smooth in texture. If not using a food processor, chop the fruit into small pieces (less than ¼ inch pieces).
Using an electric mixer, cream together the butter, sugar, and lemon zest.
Add the olive oil and mix until just combined.
Add the eggs one at a time, mixing until combined. Be careful not to overmix.
Add the lemon mixture to the butter and egg mixture and mix until just combined.
In a separate bowl, whisk together the flour, baking powder, and baking soda.
Add the flour mixture to the egg mixture and mix until combined.
Bake the cake for 55 minutes or until the top of the cake has darkened slightly and a toothpick inserted into the middle emerges cleanly.
Allow the cake to cool for 15-20 minutes before removing it from the pan.