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Q & A: Dimension of the Logarithm's argument

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Most recent answer: 02/16/2012
Q:
I want to know that whether the argument of the Logarithmic function need to be dimentionless or not? I am told that argument of Log function may or may not be dimentionless. But I am confused about it. Also when I am taking Log of "1 cm" then what will be its dimention?
- sudesh
roorkee,uttrakhand,India
A:
In order to make sense, the argument should be dimensionless.  In engineering the argument of the logarithm is always a ratio of two quantities having  the same dimensions.   In mathematics the argument of a logarithm function is simply a number.  No dimensions involved.

LeeH

But sometimes people sloppily write expressions involving logs of numbers with dimensions. Those can't actually be evaluated. If you ask how the expression changes with changes in the argument, you can evaluate that because the change involves the log of the ratio of two numbers whose dimensions cancel. Mike W.

(published on 10/22/2007)

Follow-Up #1: dimensional analysis and logarithms

Q:
Dear Ask the Van, The link above is actually a perfectly fair answer to the question asked. However, as a fellow physicist (who has since changed fields), I wanted to draw your attention to the fact that two of my more deluded colleagues in the field of ecological economics are misquoting that answer in a recent response to a critique of their article. This response can be found here: http://www.sciencedirect.com/science/article/pii/S0921800912000390 The point of their original article is that one can't ever ever take the log of dimensional variables. Of course, as all reasonable people agree, this is completely standard practice in science, and the only issue is how the quantity or analysis are modified if the units are changed (i.e. log (1 [cm]) vs. log (0.01 [m]). I just thought you might want to be aware of the extent to which that answer was misquoted, especially since it is now in the peer-reviewed literature record. Sincerely, Julia Steinberger
- Julia Steinberger (age 37)
Leeds, UK
A:
Julia- Thanks (sort of) for bringing this dispute to our attention. I just did a first skim of the paper whose URL you provided and of the paper it was critiquing.  I haven't yet noticed the point where we get quoted. My first general impression is that the arguments on both sides are far too long and complicated. We take a sort of common-sense practical intermediate position.

So let me try to make our view clear for all your colleagues. It is sloppy to use dimensional quantities in the argument of a log function. In real life, we all get sloppy sometimes yet still survive. So long as there's a point at which the difference of two such sloppy logs will be taken, as in a time derivative, no harm is done. We get away with precisely such slop in elementary statistical physics. (See Boltzmann's tomb.) Sometimes even an implicit understanding of what the natural units are might make it sort of ok. Other times one is left with useless pseudo-equations.

BTW, it was interesting to see it mentioned that there are papers going back to at least 1960 in which people noted that Marx's comparisons of rates of income and amounts of capital are dimensional nonsense. I'd wondered if anybody else had noticed.  That would be an example in which repairing the equations to the point where the argument could even be evaluated would require adding some whole new line of argument, e.g. setting some characteristic time scale such as a human lifetime.

So again, there are points where dimensional errors make arguments worthless and other points where they are really only temporary lapses in notation on the way to a sound argument. Your colleagues don't seem to be "seriously deluded" but rather seem to be slightly over-reacting to some of the real errors that have been made.

Mike W.

(published on 02/16/2012)

Follow-Up #2: more on dimensional analysis

Q:
Hi Mike W. , Thanks for your answer, and you're right, I messed up my link in my earlier note, it should have been http://www.sciencedirect.com/science/article/pii/S0921800912000419 "Response to “dimensions and logarithmic function in economics: A comment”" by Mayumi & Giampietro You should be able to find the place you are (mis)quoted very easily now. In terms of your answer, I would go further: using dimensional variables in logs (and all other kinds of non-linear functions) is fine as long as the implications of unit change can be understood (i.e. which parameters and constants change, and how). When a physicist takes log(a), where a is in centimeters, it is always understood that it is shorthand for log(a [cm]/1 [cm]), or more generally log(a [units]/1 [units]), so the number inside the log is really dimensionless, and everything related to the dimensionality has to be dealt with in other parts of the analysis. It's just that this would be tedious to write out all the time ... Anyway, thanks if you can change the post to reflect the real link. The first link I mistakenly sent you is actually a very reasonable response to the original Mayumi & Giampietro article. Sincerely, Julia
- Julia Steinberger (age 37)
Leeds, UK
A:
Hi Julia- I'm not sure I agree that it's a good idea to leave those mathematical complications implicit rather than explicit. Readers tend to fill in gaps in very surprising ways. Of course the effect of a scale change on a miscellaneous non-linear function is much messier than its effect on a log. Anyway, you have to give M&G credit for one thing- they cited us even if they didn't quite fully get what we were saying.

Totally off-topic, I liked your first link too because it had a lot of discussion of Barry Commoner. At my first teen-age lab job Barry stuck his head in the door and asked "What are you doing here?" I was honest: "Breaking things." He asked "Is someone paying you to do it?" When I said yes, he was satisfied and wandered on. Word in the halls was that his relation to math was similar to my relation to glassware.

Mike W.

(published on 02/16/2012)

Follow-up on this answer.