IP changes every time you turn off your modem/router, unless you have a static business IP (which you usually pay extra for).
ISPs keep logs of which customers use which IP at any given time (they have your home address, how else do they send you a bill), usually for about a couple of years at least (here in America anyway)
If you want to avoid being spied on, use a VPN:https://torrentfreak.com/which-vpn-providers-really-take-anonymity-seriously-111007/
only drawback is that they charge for it, but he.y, you do what you gotta do. maybe share an account with a few friends etc
That said, it is only a matter of time that they ban VPNs either on a federal level or on a private level. Many people have been trying to guess when it will happen: https://torrentfreak.com/how-long-before-vpns-become-illegal-120615/
Case in point, ebay's Paypal has recently banned a VPN provider: https://torrentfreak.com/paypal-bans-bittorrent-friendly-vpn-provider-120622/
You have to keep in mind that there has always been spying, in particularly public torrents. Usually it is hired 3rd party scouts, that the Music and Hollywood biz hire to track down and in turn they send a letter to the ISPs to look into it (You are easily traceable via public torrents: http://youhavedownloaded.com
companies set up spiders [kinda like google but for torrents] to search public torrent sites and the seed/peers/leechers using them). In turn, the ISP sends you a cease and desist letter. Sometimes it is mostly done to scare you (people usually just ignore them), others have actually turned out be lawsuits. The RIAA has even sued dead people: http://arstechnica.com/uncategorized/2005/02/4587-2/ "In 2003, they mistakenly targeted a 66-year-old woman for allegedly sharing gangsta rap. But this case goes a bit further, as Mrs. Walton actually passed away in December 2004."
That said, ISPs themselves have usually stayed neutral in the sense that it only acted upon with evidence provided to them or a warrant or w/e. So them making deals with the MAFIAA privately is pretty creepy. Only time will tell how effective and how legal it all is.
There are other ways to stay anonymous online, such as the Tor project (https://www.torproject.org/
), but that isn't recommended for torrents at least, due to limited bandwidth. Usually people dealing with sensitive info use Tor e.g. activists in potential danger, journalist, military or uhh even the FBI/law enforcement uses it:https://www.torproject.org/about/torusers.html.en