A digital zoom is not a true zoom - it does not produce the same result as an optical zoom lens.
An optical zoom lens changes the perspective. By moving the elements of a lens it brings the subject closer to or further away from the photographer without any significant loss of image quality.
A digital zoom achieves a similar effect digitally by cropping the image in camera. However, it does not change the perspective, therefore the increased telephoto effect is an illusion. Also by cropping the image it reduces the number of pixels which at the extreme 'telephoto' will cause pixelation as well as increasing the risk of camera shake.
Most of what a digital zoom does can be achieved with a computer using an Image Maniplation programme like Photoshop or Paintshop Pro. The advantage is that provided the photographer makes a copy first, any adjustment to the image can be undone, something that cannot be achieved with a digital zoom in camera.
Treat with caution!

Digital Zoom