All the current proposals for a space elevator seem to be for one to the GSO.However is there some proposal for a space elevator to a low earth orbit? I know that it would require continuous power to just maintain it in a synchronized position, but it still seems to be worth it.
The reason it's hard to get to orbit isn't that space is high up.
It's hard to get to orbit because you have to go so fast.
Space isn't like this:
Space is like this:
Any explanation I would produce cannot be possibly better, so I shamelessly copied it from the canonical source https://what-if.xkcd.com/58/
The answers to the existing question "What is a space elevator?" also answer your question; they explain that elevator needs to go all the way up to the GEO and beyond.
While in a technical sense a space elevator to LEO can't exist because it would just fall down there is another option called a rotovator--and it's much easier to build, besides.
You take a long cable and put it in orbit, the whole cable is spinning on the short axis. You orient this so the end facing the planet is on the backwards part of it's spin. Note that the tip of this cable will be moving far below orbital speed, for the biggest rotovator it can even come to a complete stop. (But picking something up off the ground isn't practical for safety reasons.)
Unfortunately, the rotovator has two weaknesses:
1) You have to dock with it on the fly, you can't just board it like you can an elevator.
2) An elevator is in continual outward tension, so long as the weight in transit is less than the outward tension it's fine. It's like a printing press--continually turning out mass in orbit. A rotovator isn't anchored, it has zero outward tension. Everything you lift lowers the rotovator. Thus it functions like a bank--you can only withdraw what you put in. The only reason this is of any use is that what you put in and what you take out need not be the same thing. You can boost spacecraft and deorbit trash.
From the linked question in kubanczyk's answer you can understand the principle of operation of a space elevator. As to what to do about getting to another orbit such as LEO you have a choice.
As an example, if you had a "conventional" supersynch elevator and released something in a high MEO it would not enter a circular MEO orbit but an ellipse with its apogee in MEO and perigee at some lower point in LEO. You'd have to work out the target MEO release point in order to minimise the delta-V to attain your desired LEO orbit. This might be large, e.g. > 1km/s, depending very much upon your definition of "LEO", but you have at least sidestepped the ~9.5km/s to go straight to LEO from the surface.
By "conventional supersynch elevator" I mean that, i.e. one that terminates well beyond GSO and is balanced appropriately, see geoffc's answer to the other question and the comments after it. Its entirely up to the user where on the structure to release an object.
Thought experiment: If it is released at GSO altitude then it will adopt a circular orbit. If it is released slightly sub-GSO then it will have a GSO velocity at an altitude for which a circular orbit would have to have a higher velocity. As a result it will adopt an elliptical path with a lower perigee. Keep on lowering the release point and explore what happens to the perigee.
There are some helpful explanations and diagrams in CoAstroGeek's answer to this question Needed height from the release of a space elevator to get a stable elliptical orbit? and HopDavid's answer to this one If we build a space elevator from Earth surface to GEO, could I step off it at GEO and remain in GEO?