Say if I have two Areas of size
a = 6 km$^2$ and b = 0.1 km$^2$
how would I find how many orders of magnitude one is greater or less than the other.
So, the way I would work this is by saying that 0.1*10 = 1 and 1*10 = 10, making a roughly 2 orders of magnitude greater than b. Is this correct? What if I wanted to calculate the order of magnitude difference between 5.7 and 6.7? What would that equal?
Also, if both locations emit a radiation of a_Rad = 1.5, and b_Rad = 3, i.e one is half the other, is it correct to then say that the difference in radiation is related to the difference in area, where the relationship in radiation difference is driven by the order of magnitude difference between the area?
Best Answer
As you say*"order of magnitude" is usually taken to mean powers of ten. It is not exact. A reasonable approach is to take the difference of the base $10$ logs, then round to the nearest integer. So your example of $6$ vs $0.1$ the difference of logs is $0.778-(-1)$, which rounds to $2$ as you suggest. On this scale, $5.7$ and $6.7$ are the same order of magnitude.
Douglas Hofstadter had an interesting Scientific American column suggesting that for small numbers (certainly up to 100, maybe 10,000) you notice individual objects, then you notice the log, maybe up to $10^{20}$ or so, then you notice the log of the log, and so on.