B737 FLT CREW REPORTS FAILURE TO CALL LAX FOR LANDING CLEARANCE BUT RECEIVES GREEN LIGHT FROM TOWER ON SHORT FINAL.

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DSNDING INTO LAX WE WERE ASKED TO KEEP OUR SPD AT 250 KTS AS LONG AS POSSIBLE FOR TFC FOLLOWING US TO RWY 25L. AS WE STARTED TO DIRTY UP; APCH TOLD US TO SWITCH TWR TO LIMMA. I SAW THE FO DIAL IN THE TWR FREQ; AND RETURNED BY CONCENTRATION TO THE APCH. RADIO TFC WAS BUSY AND I THOUGHT I HEARD CLRED TO LAND. AT ABOUT 300 FT AGL I NOTICED THE ALDIS LAMP SHINING GREEN FROM THE TWR. NORMALLY I USE THE OUTBOARD AND PULSE LIGHT (-300) AS A VISUAL REMINDER OF OUR LNDG CLRNC. THIS -300 HAD AN MEL ON THE OUTBOARD LIGHTS WITH A NOTE NOT TO EXTEND THEM SO I HAD NOT TURNED THEM ON WITH LNDG CLRNC. MY THOUGHT WAS THE ALDIS LAMP WAS TRAINING IN PROGRESS AS I WAS SURE WE WERE CLRED TO LAND. AFTER WE CLRED BTWN THE RWYS; TWR WAS SLOW TO GIVE US OUR HOLD SHORT INSTRUCTIONS. I STOPPED THE ACFT AND LOOKED DOWN TOWARDS THE RADIO JUST AS APCH ASKED US IF WE WERE STILL UP THEIR FREQ. AT THIS POINT WE REALIZED OUR ERROR AND THE REASON FOR THE VISUAL LNDG CLRNC. TASK SATURATION AND A DESIRE TO HELP OUT PULLED ME OUT OF MY NORMAL ROUTINE AND; WERE IT NOT FOR A QUICK THINKING TWR CTLR; WE WOULD HAVE LANDED WITHOUT CLRNC. I CALLED TWR TO THANK THEM FOR THEIR HELP AND APOLOGIZE FOR OUR ERROR. I FEEL FORTUNATE NOT TO HAVE BEEN IN A WORSE SIT. TASK SATURATION AND FAILURE TO VERIFY. TEXTBOOK ERRORS THAT WE REVIEW ALL THE TIME. TRUST BUT VERIFY AND DON'T BREAK GOOD HABIT PATTERNS OUT OF A DESIRE TO HELP. SUPPLEMENTAL INFO FROM ACN 792942: I REALIZE A LOT WAS GOING ON; I WAS CLRING TFC; MONITORING THE APCH; WORKING THE RADIOS; AND I ALLOWED MY XCHK TO SLIP UP. I OVER CONCENTRATED ON THE SLOWING TO CONFIGURE AND THE CALLOUTS AND FORGOT TO CHK IN WITH THE TWR. TWR WAS QUICK THINKING IN USING THE GREEN LIGHT. I HAVE LEARNED A VERY VALUABLE LESSON AND WILL PRIORITIZE BETTER IN MY PLT MONITORING DUTIES IN THE FUTURE. PRIORITIZE COCKPIT DUTIES. ENSURE CHKING IN WITH TWR WHILE OPERATING IN THE DYNAMIC APCH ENVIRONMENT.