The wikipedia article was actually pretty interesting on this point. They suggest that 10 is commonly used, but other bases may be contextually relevant.

> An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually 10

I guess you could think of it like a _really_ low precision float or something.

1: https://en.wikipedia.org/wiki/Order_of_magnitude

I suspect the phrase is a cliche often used to sound scientific and sometimes by folks unaware, like how description of growth as “exponential” is a cliche used by non-mathematical discussion.

I agree with this. As a layman, I've always understood "orders of magnitude larger" to just mean "way too big" and "exponential" growth to imply "out of control".

x^2 and 2^x manifestly both involve exponents, so I think it's valid - outside math class - to call anything involving accelerating growth "exponential".

I usually interpret it as "scales geometrically and not arithmetically".

That seems confusing. If I say my new service is orders of magnitude more efficient than previous services, I don’t mean any thing about scaling but current performance, and I wouldn’t say orders of magnitude if it was just twice as many calls/second/core, but more than ten, or really more than 30, halfway between ten and a hundred, logarithmically.

Something that scales geometrically might well have some giant constant so it isn’t useful until a specific performance regime.

i think of "orders of magnitude" to mean "powers of ten", but I asked since i'm not sure i'm reading what folks think they are saying.

I interpret it as saying: exponentially different, where the range in my uncertainty is comparable tonthe effect of the choice of base.

For instance if the range is '3-13' in base 2 it's similar to '1-4' in base 10, but either way I'm making up numbers so who cares what the base is.

Base 10 is certainly the default. Decibels are base 10; more specifically a Bel is 1 factor of 10, and a decibel is 1 tenth of that, ie 10^0.1.

fractional exponents don't work like that, for example x^½ is the square root of x. You probably meant 10^-1 !

No, 1 decibel is 10^1/10. Bels are multiplicative, not additive. You multiply 10^.1 by itself 10 times and you get 10^1. Similarly if you multiply x^1/2 by x^1/2 you get x.

> when people say "orders of magnitude" do they always mean base 10?

Yes, and that has always bothered me just a little bit, given that there is nothing intrinsically special about base 10 and yet the phrase seems to suggest something fundamental.