I don’t know of any good explanations; this seems relevant but requires a subscription to access. Unfortunately, no-one’s ever explained this to me either, so I’ve had to figure it out by myself.
What I’d add to the discussion you linked to is that in actual practice, logarithms appear in equations with units in them when you solve differential equations, and ultimately when you take integrals. In the simplest case, when we’re integrating 1/x, x can have any units whatsoever. However, if you have bounds A and B, you’ll get log(B) - log(A), which can be rewritten as log(B/A). There’s no way A and B can have different units, so B/A will be dimensionless.
Of course, often people are sloppy and will just keep doing things with log(B) and log(A), even though these don’t make sense by themselves. This is perfectly all right because the logs will have to cancel eventually. In fact, at this point, it’s even okay to drop the units on A and B, because log(10 ft) - log(5 ft) and log(10 m) - log(5 m) represent the same quantity.
I don’t know of any good explanations; this seems relevant but requires a subscription to access.
Most of that paper is the authors rebutting what other people have said about the issue, but there are two bits that try to explain why one can’t take logs of dimensional things.
Page 68 notes that
y=logbxifx=by, which “precludes the association of any physical dimension to any of the three variables b, x, and y”.
And on pages 69-70:
The reason for the necessity of including only dimensionless real numbers in the
arguments of transcendental function is not due to the [alleged] dimensional nonhomogeneity of the Taylor expansion, but rather to the lack of physical meaning of including dimensions and units in the arguments of these function. This distinction must be clearly made to students of physical sciences early in their undergraduate education.
That second snippet is too vague for me. But I’m still thinking about the first one.
The (say) real sine function is defined such that its domain and codomain are (subsets of) the reals. The reals are usually characterized as the complete ordered field. I have never come across units that—taken alone—satisfy the axioms of a complete ordered field, and having several units introduces problems such as how we would impose a meaningful order. So a sine function over unit-ed quantities is sufficiently non-obvious as to require a clarification of what would be meant by sin($1). For example—switching over now to logarithms—if we treat $1 as the real multiplicative identity (i.e. the real number, unity) unit-multiplied by the unit $, and extrapolate one of the fundamental properties of logarithms—that log(ab)=loga+logb, we find that log($1)=log($)+log(1)=log($) (assuming we keep that log(1)=0). How are we to interpret log($)? Moreover, log($^2)=2log($). So if I log the square of a dollar, I obtain twice the log of a dollar. How are we to interpret this in the above context of utility? Or an example from trigonometric functions: One characterization of the cosine and sine stipulates that cos^2+sin^2=1, so we would have that cos^2($1)+sin^2($1)=1. If this is the real unity, does this mean that the cosine function on dollars outputs a real number? Or if the RHS is $1, does this mean that the cosine function on dollars outputs a dollar^(1/2) value? Then consider that double, triple, etc. angles in the standard cosine function can be written as polynomials in the single-angle cosine. How would this translate?
So this is a case where the ‘burden of meaningfulness’ lies with proposing a meaningful interpretation (which now seems rather difficult), even though at first it seems obvious that there is a single reasonable way forward. The context of the functions needs to be considered; the sine function originated with plane geometry and was extended to the reals and then the complex numbers. Each of these was motivated by an (analytic) continuation into a bigger ‘domain’ that fit perfectly with existing understanding of that bigger domain; this doesn’t seem to be the case here.
How are we to interpret [the logarithm of one dollar] in the above context of utility?
You pick an arbitrary constant A of dimension “amount of money”, and use log(x/A) as an utility function. Changing A amounts to adding a constant to the utility (and changing the base of the logarithms amounts to multiplying it by a constant), which doesn’t affect expected utility maximization. EDIT: And once it’s clear that the choice of A is immaterial, you can abuse notation and just write “log(x)”, as Kindly says.
I don’t know of any good explanations; this seems relevant but requires a subscription to access. Unfortunately, no-one’s ever explained this to me either, so I’ve had to figure it out by myself.
What I’d add to the discussion you linked to is that in actual practice, logarithms appear in equations with units in them when you solve differential equations, and ultimately when you take integrals. In the simplest case, when we’re integrating 1/x, x can have any units whatsoever. However, if you have bounds A and B, you’ll get log(B) - log(A), which can be rewritten as log(B/A). There’s no way A and B can have different units, so B/A will be dimensionless.
Of course, often people are sloppy and will just keep doing things with log(B) and log(A), even though these don’t make sense by themselves. This is perfectly all right because the logs will have to cancel eventually. In fact, at this point, it’s even okay to drop the units on A and B, because log(10 ft) - log(5 ft) and log(10 m) - log(5 m) represent the same quantity.
Most of that paper is the authors rebutting what other people have said about the issue, but there are two bits that try to explain why one can’t take logs of dimensional things.
Page 68 notes that y=logbx if x=by, which “precludes the association of any physical dimension to any of the three variables b, x, and y”.
And on pages 69-70:
That second snippet is too vague for me. But I’m still thinking about the first one.
[Edited to fix the LaTeX.]
The (say) real sine function is defined such that its domain and codomain are (subsets of) the reals. The reals are usually characterized as the complete ordered field. I have never come across units that—taken alone—satisfy the axioms of a complete ordered field, and having several units introduces problems such as how we would impose a meaningful order. So a sine function over unit-ed quantities is sufficiently non-obvious as to require a clarification of what would be meant by sin($1). For example—switching over now to logarithms—if we treat $1 as the real multiplicative identity (i.e. the real number, unity) unit-multiplied by the unit $, and extrapolate one of the fundamental properties of logarithms—that log(ab)=loga+logb, we find that log($1)=log($)+log(1)=log($) (assuming we keep that log(1)=0). How are we to interpret log($)? Moreover, log($^2)=2log($). So if I log the square of a dollar, I obtain twice the log of a dollar. How are we to interpret this in the above context of utility? Or an example from trigonometric functions: One characterization of the cosine and sine stipulates that cos^2+sin^2=1, so we would have that cos^2($1)+sin^2($1)=1. If this is the real unity, does this mean that the cosine function on dollars outputs a real number? Or if the RHS is $1, does this mean that the cosine function on dollars outputs a dollar^(1/2) value? Then consider that double, triple, etc. angles in the standard cosine function can be written as polynomials in the single-angle cosine. How would this translate?
So this is a case where the ‘burden of meaningfulness’ lies with proposing a meaningful interpretation (which now seems rather difficult), even though at first it seems obvious that there is a single reasonable way forward. The context of the functions needs to be considered; the sine function originated with plane geometry and was extended to the reals and then the complex numbers. Each of these was motivated by an (analytic) continuation into a bigger ‘domain’ that fit perfectly with existing understanding of that bigger domain; this doesn’t seem to be the case here.
You pick an arbitrary constant A of dimension “amount of money”, and use log(x/A) as an utility function. Changing A amounts to adding a constant to the utility (and changing the base of the logarithms amounts to multiplying it by a constant), which doesn’t affect expected utility maximization. EDIT: And once it’s clear that the choice of A is immaterial, you can abuse notation and just write “log(x)”, as Kindly says.