The smaller you make the camera sensor, so the shorter focal length you need for any given angle of view.
APS or 'Crop' sensor SLR's have a sensor aproximately half the area of a full-frame or 35mm SLR. It's roughly 24mm wide by 16mm tall, instead of 36mm wide by 24mm tall.
We now get into 'similar triangles; and 36mm/24mm or 24/16 = 1.5, the effective 'Crop-Factor'.
So, by this similar triangles crop factor; the angle of view of a 50mm lens on a full frame camera is the same of that of a 33mm lens on a Crop sensor. Ie you need less lens length to get the same 'zoom'.
Now make the sensor even smaller; I have a little Kodak compact to hand; with 3x Zoom that proclaims 35mm-105mm 'equivilent' on the front. Exif data says it has an actual lens length of 5.8mm - 17.4mm.... a zoom range of 3x.
Hmm... this is where it gets a bit, err.. deceptive. That 'Zoom Times' doesn't actually tell you how much magnification the lens gives....its just the ratio of longest lens length to longest!
Make the shortest lens length very short, you don't need to go very long to get a very big zoom-times, do you?
I mean, on a 35mm camera, 10mm is pretty wide, probably a fish-eye.. but if I made a 10-100 zoom, I could claim 10x zoom factor... but, gives no more far subject maginifcation than little digi-compact that has only 3x Zoom, but starting from a less wide angle at the small end.
However; the huge crop factor of a micro-sensor compact means you dont need very long lenses to get big 'zoom'. Just 17.4mm on that Kodak, gives the equivilent of 105mm on a 35mm camera. 17.5 on my Crop Sensor DSLR is still a wide angle lens, equivilent to about 26mm on 35mm.
So a 30x zoom? Doesn't actually tell you much; but lets say that they used some of it to get a wider angle of view than 35mm Full-Frame equivilent... and started at something like; 24mm equivilent... we have a 6x crop factor, so that would need a 4mm lens, on compact sensor. 30x 4mm is 120mm; which isn't all that long, and would only be a mild tele-photo on a 35mm camera but with 6x crop factor, that has the effective magnification of a 720mm super-tele on 35mm.
YET the zoom range in terms of focal length is smaller than that of a 70-210 3x soom on a 35mm camera. That gives 180mm of extension, this 30x zoom is giving only 115...
Its magnification rating is coming fro a HUGE crop factor, curtecy of a tiny tiny sensor.
Digital zoom? Basically doubling that again, by simply only taking data of the central 1/3 or so of the already tiny sensor, increasing the crop factor from 6x to 12 times, and doubling again the 'effective' zoom range and magnification.
Great if you want lots and lots of telescopic reach; but 450mm on a full frame camera is enough to fill the frame with the moon.... how much magnification do you want?
Its 'Cheap' magnification; and its at the expense of image quality.
Digital Zoom is no better now than it was ten years ago, and its still doing the same thing, the same way; halving the utilised sensor size, to double the crop factor, and double the effective lens length; but obviousely, if you have a 10x or 20x or 30x optical zoom, doubling that by digital cropping, is going to give much bigger effective numbers than if you had a 3x or 5x or 7x zoom....
Ah! but more modern cameras probably have twice the Pixel Count of older ones!
Which is why they can 'get away' with it! As many pixels in that cropped 2x zoom on modern camera, as there was on early one using the full frame...
However... the sensor is no bigger... the sensor is getting no more light... theres still the same number of photons falling on the sensor, so each pixel is getting less of them, so more 'amplification' is needed to get a signal from them.... means more noise and lower quality.
All in all, its a BIT of a con.... and when it comes to cameras, the numbers are usually always ratios, not actual values... you have to know what they are doing and how they work and relate to each other to know whether they are any more of less useful.... but the marketing men like the big numbers they can present as 'impressive' on the consumer market... in cameras they make down to an 'Acceptable Quality Level' both for price and performance.
And they can get these big impressive 'specs' very easily just by using a smaller sensor area. THAT is all it is. Inflation by crop-factor. Confusion by Pythagoras!