Preheat your oven to 350°F (175°C).
In a mixing bowl, cream together the butter and sugar until light and fluffy.
Add the eggs and vanilla extract, and mix until combined.
Gradually add the flour and baking powder, mixing until just combined.
Press the cookie dough into the bottom of a greased baking pan.
In another bowl, beat the cream cheese and sugar until smooth.
Add the eggs and vanilla extract, and mix until combined.
Pour the cheesecake mixture over the cookie base in the baking pan.
Bake for 35 minutes, or until the cheesecake layer is set.
Allow to cool completely before cutting into bars.