His father confirmed it.
L when tells Light his real identity, he made it in such a way that Light couldn't kill him.
- If Light tries to kill "Hideki Ryuga", he is not sure if he is the real L. If the fake imposter dies (Like the one on TV), Light confirms he is Kira
- He might kill the real "Hideki Ryuga" by mistake and not gain anything while exposing himself.
But he plays L's game and finally ends up in the HQ of L's team where his father confirms he is the real L. Later when L realizes about the Shinigami eyes, he tells Light that he has made arrangements with his Father that if he dies in the next few days that Light is Kira.
Thus he didn't get a chance to kill L until he was able to manipulate Rem.
EDIT: Found the source. Light's father confirms this in hospital in Manga Chapter 22. Here is the screenshot.