The internal logic of a topos

As I understand the internal logic of a topos it consists of certain morphisms from finite powers of Omega to Omega. In the case of Set it consists of all such morphisms. For which toposes is this not the case, and for those how are the morphisms that do belong to the internal logic best characterized? I do hope it's not necessary to start from the notion of an internal Heyting algebra, that sounds so counter to mathematical practice and intuition. If the internal logic consists of precisely those morphisms preserved by geometric morphisms this will give me the necessary motivation to go to the mats with geometry. Vaughan