Why would the 4 year laminate wood floor completely change colors only under where there has been a rug? The area under the rug is at least 4 shades lighter than where the floor was rug free? And what can I do to make the floor be all one color again.

What type of laminate do you have is it a floating floor ?

Sorry, but I don't know what a floating floor is. I bought the house right before it was
Finished and opted for the laminate floor in lieu of carpet.

All you can do is take up the rug and let the sun bleach it out as well. This may take some time to notice if at all.

If there is no sun hitting this area then it may be the cleaner you've been using

As far as removeal of the floor in those areas it will be a job to do this, you've have to take up all the floor just to get the area and then lay it all back again,

I agree with glavine if it is a large room it could be tedious to take up and reinstall I have seen some flooring actually get darker from sun exposure instead of lighter. Just a side note a floating floor is a floor that is attached only to itself not the sub floor underneath. I would try following glavine's advice first before taking any more drastic measures.

The area underneath the rug lightened up significantly while the other area stayed the
Same. I only clean the floor with a swiffer to remove dust and use a barely damp mop
When absoloutely necessary. Makes me wonder if the binding of the underside of the rug had some effect on the floor and made it become so much lighter. I can only afford to put the old rug back down and live with it as is!! I was hoping maybe somebody could
Explain or maybe had the same happen to their floor also.

I have seen rubber backed rugs discolor floors but haven't yet seen it happen on laminate.

Try a small amount of mineral spirits on a rag to see if the color under the rugs can be removed. If you have any scraps of the flooring try the spirits there first but it should be Ok. I think AC101 is on track, the floor has darkened very slowly over 4 years so you might not really remember what it originally looked like. Some floors dark with UV and age, kind of like us but without the wrinkles.