Hey I have a 2010 Mazda 3 I use the 5w20 synthetic, how many miles should i go between oil changes?

Thanks

 

3000k from brand new, then every 5000k. thats what i do.

 

I do every 5000 with dyno oil (mobil 5000) - have never used synthetic. The synthetic should run longer, but i'd suggest having someone run an oil analysis every few thousand to see how long it can go... Or until it looks dirty.
There is no suggestion from mazda about using synth, so i'd be weary just in case you have any engine issues they could easily claim that you're not following the maintenance schedule.