Did Light knew L's real name? In Death Note, Light picked up Rem's notebook after he/she died by writing L's name his/her the notebook

Does it mean that Light now knew L's real name?
Maybe he did not bother checking his rival's name or something...i guess he saw the name but maybe it was of no significance to him...and also in one of the early episodes light says that he is terrible with names and faces..so most probably he saw the name but forgot it or he saw it but it was not revealed to us whether he now knew  his actual name or not or either he just simply didn't bother to check his name...