Just to be clear, you're not losing anything when you use an FX lens on a DX body.
In days of 35mm SLR cameras, the "image circle" projected by the lens into the sensor body had to be large enough to cover the size of the 35mm negative frame with adequate optical quality (the image is usually a bit larger but quickly degrades at the edges.)
When digital came along, the most expensive component in the camera was that digital sensor. It was VERY expensive. Camera makers hit upon the idea that they could substantially lower the cost by producing a camera body that used a sensor with a reduced physical size, but placed inside a body that could still use all the 35mm lenses. Basically that means the sensor is only capturing the part of the image near the center of the lens' "image circle" -- it's as if you took a full 35mm frame but then cropped away the outer edges of the image, then enlarged the image to print the same size. Hence the name "crop frame" sensor. The size they chose happened to be nearly the same size as the Advanced Photo System "Classic" film negative size (that's where the acronym "APS-C" comes from.) This massively reduced the expense of the camera and allowed for compatibility with regular lenses.
DSLRs were also eventually produced with sensors which were the same size 35mm film camera negatives, but these were extremely expensive (they still are quite a bit more expensive... an "entry" level full-frame body is just slightly less than $2000 for the body-only).
So where do DX lenses (on Canon they're called EF-S) fit into the story?
It was realized that since the APS-C size DSLRs aren't capturing the full size image, that you could save a lot of money by building lenses that didn't project a large image circle. Also, the optics only need to be good enough to hold image quality out to the corners of a much smaller sensor size. This means the elements can be smaller, they're easier to grind to precision, the lens can be more compact and it will weight less. And (this is the best part) there's absolutely NO loss of image quality. The image quality is every bit as good. The only difference is that the lens doesn't project an "image circle" into the sensor body large enough to reach the corners of a full-frame sensor (that is, a sensor which is as large as a 35mm film negative.)
If you buy an FX lens, you may use it on EITHER a DX or FX body. If you buy a DX lens, it will really only be suitable on a DX body... only because it cannot fill the frame of an FX body.
If you want to own a 70-200mm lens, do not let the fact that you have a DX body stop you. The lens will work flawlessly on your camera.
Also.. there is no difference in focal length between FX and DX. The focal lengths of the lenses are always stated as "true" focal lengths -- not "effective" focal lengths (as are often stated on point & shoot cameras). A 100mm focal length on a DX body and a 100mm focal length on an FX body are the same... the difference is only that in a DX lens, the image spills off the sensor and only the middle of the image is captured. This creates the illusion that you've "zoomed in" an extra 50% (that's the Nikon crop-factor for APS-C) on a DX body (so it "seems" like you're using a 150mm focal length -- but of course if you took an FX image and cropped in a tighter you'd get EXACTLY the same effect.)