Scan resolution provides enlargement. The ratio of (scanning resolution / printing resolution) is the enlargement factor.
For example, if printing:
Scan at 600 dpi, print at 300 dpi, for 600/300 = 2X size (double size or 200% size)
Scan at 300 dpi, print at 300 dpi, for 300/300 = 1X size (original size or 100% size)
Scan at 150 dpi, print at 300 dpi, for 150/300 = 1/2X size (half size or 50% size)
Film is small, and needs more enlargement. Prints have already been enlarged.
One example is to scan 35 mm film at 2700 dpi, and print at 300 dpi, for 2700/300 = 9X size enlargement. 9X is about 8x12 inches (about A4 size) from full frame 35 mm (about 0.9 x 1.4 inches).
The ratio of (scanning resolution / printing resolution) is the enlargement factor.
This is called "scaling", and this enlargement concept is true for scanning anything, photo prints, film, documents, etc.
For example, for the video monitor screen:
Scanning 4 inches of paper
at 100 dpi gives an image size dimension of 4x100 = 400 pixels.
at 200 dpi gives an image size dimension of 4x200 = 800 pixels.
at 300 dpi gives an image size dimension of 4x300 = 1200 pixels.
Scanning 35 mm film size at 2700 dpi creates enough pixels to print 8x10 inches at 300 dpi (and allow a bit extra size for composition cropping). The film and the print are different aspect ratios (different shapes), so a bit of cropping of the original is necessary to fit the paper shape (see Cropping, Resampling, Scaling).
This Input size and Output size being the same pixels is the numeric concept of printing enlargement. Film enlarges well, however enlarging copies of photo prints much larger than original size becomes a quality problem.
Scanning film needs greatly more resolution than is practical if scanning prints. Because, film is generally small, and needs greater enlargement. Prints have already been enlarged when created. And film is a higher resolution fine grained media, designed to plan for this enlargement. Prints are not, instead designed only to be "good enough" when viewed by human eye.
When we get right down to it, scanning color prints can rarely yield more detail when scanned at more than 300 dpi. I am carefully saying color prints, to exclude film and even B&W prints. In particular, I'm specifically speaking of typical 6x4 inch (4X size) 35 mm color photo prints from the regular photo-finisher. Not just any "image", certainly not one printed in a magazine or book, but scanning a real photo print on actual photo quality print paper. Yes, the original negative or the original full sized camera digital image file have fantastic resolution, a full order of magnitude better than the copies of prints. The photo prints, well, they are a lesser copy of that film original.
Prints do vary, some are simply sharper than others. The camera lens is quite good, although an inexpensive camera that can't be focused is not the best performer. If the camera were on a steady tripod, it could deliver sharper images than the shaky results when we handhold them. Faster shutter speeds help. These factors influence the sharpness of the original image on film, and not all images are equal. Then, print size enlargement specifically hurts. Enlarging a film image by a factor of 4 or 8 reduces resolution by a factor of 4 or 8. But mainly, the printing paper itself is far from the equal of the film, it is designed for a different purpose, which does not include additional size enlargement. Good film might resolve detail at 3000 dpi, but color print paper doesn't come close.
There are exceptions to this 300 dpi comment. When scanning 35 mm color photo prints of B&W resolution test targets taken with a good lens on a tripod, 600 dpi scans can sometimes show slightly more resolving power than 300 dpi scans. The results are not nearly double 300 dpi, but perhaps the absolute number is near 400 dpi from good sharp prints. Professional printing from large sheet film is a similar mild exception (much less enlargement). But we typically don't do either, and when comparing scans of real 35 mm color photos of real subjects, it is generally difficult to detect much practical benefit when scanning above 300 dpi (and frankly, sometimes above 200 dpi for unsharp snapshots from inexpensive cameras).
These exceptions suggest that in extreme cases, we should try heroic measures. I don't disagree with that. If you need to enlarge a print to triple size, the heroics are the best hope. But I am suggesting that the real life benefit of scanning 35 mm color prints above 300 dpi is mostly wishful thinking. I am trying to influence your expectations, more than your procedure. If it is easy, sure, go for it, but don't expect too much.
However, you should also know that scanning the 35 mm negative with a good film scanner is a whole different world, and then sharp detailed full page prints become easy. And this large difference clearly shows that drastically enlarging a color print is not the best method, for the reason I am trying to explain.
Anyone that has ever done any darkroom work knows it's not so easy to create a really sharp 8x10 inch photographic print from 35 mm film. To scan a photo and enlarge it to print at 8x10 on an inkjet is certainly not going to do any better. Did you ever try to copy a wallet size photo by photographing the small photo print, and then enlarging that film copy to 8x10? If so, you know its detail will be poor as compared to printing from the original negative at 8x10 (and the point is that scanning to enlarge this small print won't be different). Prints just don't contain the detail to enlarge well. Professional prepress insists on film, not prints. When scanning 35 mm color prints, you will often look very hard for detectable improvement when scanning over 300 dpi, and often that number is 200 dpi. This concept of color prints offends many that don't want to hear this, those of the "more is never enough" school, but this is very easy to test.
Notice again that I am trying to be careful to always say "color prints". B&W prints have only one emulsion layer, and might sometimes yield more detail when scanned above 300 dpi. That is nothing that I care to debate, but in particular, old historic B&W prints were often contact prints from large negatives (no enlargement at all), which may yield more detail when scanned up at 400 dpi or more, assuming size is needed.
But enlarged 35 mm color photo prints effectively stop yielding additional detail when scanned at more than about 300 dpi. This is a problem when trying to scale them larger, to still have sufficient detail to print well. The only reason I dwell on this so much is because everyone wants to scan a 4x5 inch color print, and print it to be 8x10 inch photo quality, and there is this complication involved. You can do it, and it is good up to a point, but your greatly enlarged image won't have all of the detail you might desire or imagine, and I am trying to explain why.
Certainly film can be scanned at 3000 dpi with good effect, because film and film grain were designed to be greatly enlarged. In fact, the only purpose of film is to be enlarged (perhaps a few exceptions, x-rays maybe, but certainly true in general). But photo prints and print grain were not designed to be enlarged. Prints are designed to be viewed by human eye without optical magnification. Perhaps they could have designed print paper to have finer grain and use milder developers, to support greater detail, but they did not, our human eye is not that good. I'm sure there seemed to be no reason to consider it, since the film negative original is already designed for this purpose. Some of us discard the negatives, and that's a major mistake. The film is the original master version of the image, and the print is a relatively poor copy.
This is not just my own opinion about the maximum resolution from color prints, it's a very popular opinion. For example, Doug Gennetten is the engineer with HP's Home Imaging Division doing the HP PhotoSmart printer and scanner. At the introduction of these models, Doug posted this message:
"Since printers are at or near photo quality, scanning resolution is no longer a printer-dependent value. To get photo-quality scans---even if going to a 2540 dpi image setter---200 dpi is the max resolution you need if your original is a normal silver halide color print."
The HP scanner web site says:
"A Note on Resolution: The vast majority of scanning projects require resolutions lower than 300 dpi. For example, scanning a photograph at resolutions higher than 150 to 200 dpi only produces a larger file, not more detail."
Few sources advise you this of this limitation, but knowing this can save you some fruitless headaches trying to get impossible results. Scanning film is instead the miracle you seek.
I receive a fair number of images emailed by helpful readers that say "See! Anyone can see that the 1200 dpi scan shows more detail than 200 dpi. The tiny features don't show in the smaller 200 dpi scan". I always just enlarge the smaller one and send it back, it always shows as much detail when at the same size.
Anytime you compare two resolutions for detail, you must increase the size of the smaller image to be the same size as the larger. Yes, sure, upsampling certainly degrades the smaller one, but the idea is to see if the smaller still shows all the same features and detail visible in the larger or not. If the larger one is no better, then what's the point?
A small image, so small that its detail is too tiny to be seen well, will not compare favorably with an image large enough to show that detail well. A large image enlarges detail that is too small to be seen in a smaller image, but that does NOT mean its detail is greater. You have to enlarge the smaller image to know. So before you make false conclusions about if greater detail is present or not, enlarge the smaller image to the same size, and then compare. Humans like big images, but we don't always see this objectively.
I do routinely scan color prints at 300 dpi, when appropriate for the purpose. If a large image size is needed, like for printing at enlarged size, then I do use 600 dpi, because image size is needed, and that gives size. But I don't kid myself that I get much more detail than 200 or 300 dpi will give. Still, in the chance that some major miracle might somehow occur, then I've got it. <grin>