I'm looking to get membership but I also want to see how long I've been playing RuneScape all in all.
It probably won't.
The Aventures' Log is a members-only feature, and thus does not appear if you are F2P. So your hours probably aren't counted while F2P.
If you're interested in how long you've played, talk to Hans. He will tell you how long you are close to your 5/10 year veteran cape. Then find some calculator which will figure out 10 (or 5) years minus the days Hans tells you. This will be the day you crated the account though, and won't actually tell you the amount of time you've played RuneScape.
I'm pretty sure it won't. As when I used to play this game several months ago when my membership would run out I would maybe play f2p for a couple days as I was too lazy to go to the store and buy a card.
In that small time period my log did not track my f2p progress. Even though I was only f2p for maybe 5 days I still completed a lot in game such as boss fights leveling up and finding rare drops.