1) The most obvious way of eliminating shake is to use a faster shutter speed. Either by opening the aperture wider or introducing more light by using a flash gun for instance.

Don't let the next bit confuse you, things get easier as we progress.

The rule of thumb for a sharp picture, free from the effects of camera shake, is to use a shutter speed which is at least as fast as 1 divided by the focal length of the lens. So if you are using your zoom set at 100mm you should use a shutter speed of at least 1/100 of a second. If you are using a 50mm lens you will get a sharp picture at 1/50th of a second.
The reason why the focal length of the lens is important is that camera shake becomes more apparent as the angle of view gets narrower, the narrower the angle of view the more the shake is magnified. You will know this if you have ever tried to hold a telescope or a high powered pair of binoculars still.

2) Putting the camera on a sturdy tripod is the best way to keep it still and this is the way to go, especially when you want to get some movement blur from a subject like a waterfall.

3) If you didn't take your tripod with you and you still want to take pictures in lower light situations you can stretch the rule of thumb (above) by a few stops by bracing yourself and/or the camera against a tree or wall or lamppost. Also controlling your breathing can help quite a bit too.

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