Back in the past PC Anywhere was the first choice for every remote technician. Than came similar software such as Remotely Anywhere that are easier to use and can doesn't require software to be installed in order to remote control (they have used a static HTML image map in addition to their Java Applet client). Later, services such as Microsoft Remote Assistant, Skype screen sharing (read only), logmein and TeamViewer started to provide easy access to remote machines, sometimes by proxing requests in order to make it just work without configuring port forwarding on the two sides or manually submitting IP addresses.
I've not used logmein too much, because I prefer not to work remotely on Windows computers or Windows at all, but I am not sure if the sunset on logmein free service is bad after all. Soon we will see different products and better technologies (HTML5 WebRTC anyone?), and I've not seen too much innovation in logmein in the recent years. The usage of new technologies will advance us more to the moment we could host a service similar to logmein on our own servers and provide similar service to our own or to technicians in our area.