Dual Citizen Travelling to Argentina on British Passport

Any help would be gratefully received. I keep trying to contact the Argentinian Embassy in London but I never get to speak to anyone.

I was born in Argentina and have Argentinian citizenship. When I was 5 my family moved to the UK and through my father we all obtained British citizenship. I have lived in the UK my entire life (I'm now 35) and have renewed my Argentinian passport a couple of times until I was around 20. After that I never bothered to renew my Argentinian passport.

I plan to travel to Argentina soon and was wondering if I will have any problems entering (and more importantly) leaving the country using my British passport.

I actually no longer have my Argentinian passport or identity documents so I'm not sure what the solution would be if I can't just use by British passport.