Four million is order six or order seven, depending how you round
Ask how big four million is in orders of magnitude and you can get two honest answers. Chop its base-10 logarithm, about 6.6, and you get 6. Round it and you get 7. Both conventions are in use, which says something about what orders of magnitude are for: quick, rough sense of scale, not precision.
An order of magnitude is a factor of ten. Two numbers are within an order of magnitude when the larger is less than ten times the smaller, so 1 and 9 qualify but 1 and 15 do not. Each multiplication or division by ten adds another step: 2 and 20 differ by one order, 2 and 200 by two, and 2 and 2,000,000 by six. The phrase packs a huge difference in scale into a single small number.
The simplest way to find a number's order is to count its digits and subtract one. A seven-figure income is order 6, no calculator required. Formally that equals the whole-number part of the base-10 logarithm. Rounding to the nearest order instead means rounding the logarithm, and in scientific notation the tipping point is the square root of ten, about 3.162, not 5. So 1.7 times ten to the eighth rounds to order 8, but 3.7 times ten to the eighth rounds to 9.
Order-of-magnitude estimates are prized in science as zeroth-order approximations. The number of people on Earth, for instance, lies somewhere in the band from roughly 3 to 30 billion, so its order-of-magnitude estimate is simply 10 billion. Such estimates are especially natural because many quantities in nature are spread logarithmically, making the power of ten more telling than the exact figure.
Other fields use different bases. Astronomers rank star brightness in magnitudes where each step is a factor of about 2.512, chosen so that 5 steps equal exactly 100, or two decimal orders. Computers think in powers of 2, and dedicated prefixes based on 1024 exist for electronics, while SI prefixes step in thousands. The flood of internet data has pushed the SI system to add new prefixes, most recently in 2022.
Source: Order of magnitude
