# Why are expressions such as $\operatorname{ln}(T)$ used in thermodynamics where $T$ is not dimensionless?

In all thermodynamics texts that I have seen, expressions such as $\operatorname{ln}T$ and $\operatorname{ln}S$ are used, where $T$ is temperature and $S$ is entropy, and also with other thermodynamic quantities such as volume $V$ etc. But I have always thought that this is incorrect because the arguments $x$ in expessions such as $\operatorname{ln}x$ and $e^x$ ought to be dimensionless. Indeed at undergraduate level I always tried to rewrite these expressions in the form $\operatorname{ln}\frac{T}{T_0}$. So is it correct to use expressions such as $\operatorname{ln}T$ at some level?

• You should ask at Physics. They also use $c=1$ and similar dimension-killing equations sometimes. – Hagen von Eitzen Dec 31 '12 at 14:01
• @Hagen Is it ok to post simultaneously, or should the question be moved? – user50229 Dec 31 '12 at 14:02
• Often derivatives are taken of such objects so the distinction between $\log A$ and $\log A/A_0$ is not important. – user26872 Jan 1 '13 at 3:32

You're quite right that $\ln T$ makes no sense if $T$ isn't a pure number. What physicists mean by such an expression is the logarithm of the number that expresses $T$ in some (fixed) system of units. (That's the same as $\ln\frac T{T_0}$ in your question, with $T_0$ being the chosen unit; it's also equivalent to the "drop the SI-units" in Ralph Tandetzky's answer.) Changing to a different system of units will multiply these numbers by a constant, so it will change the logarithms by an additive constant. Any fact that the physicists obtain using things like $\ln T$ will, if it has physical meaning, have to be independent of the choice of units, i.e., the additive constants in the logarithms will have to cancel out. Mother Nature doesn't know what units we want to use, and she certainly doesn't change physical reality to accommodate our choice of units. Nevertheless, the intermediate steps that the physicists use, leading up to the meaningful results, might depend on the choice of units and might therefore lead to the concerns you expressed in the question. I would advise you to think of such intermediate steps in terms of a fixed system of units (as in the second sentence of this answer), but to expect all physically meaningful results to be independent of units, and therefore only to involve such things as ratios of (absolute) temperatures or differences of their logarithms.

You're right, it's kind of odd and sloppy to write $\ln T$ and stuff like that, because you can't really tell what the logarithm of $283 K$ or $10^\circ C$ is. But if you transform all quantities into numbers multiplied with naked SI-units (without mega, kilo, milli, micro, etc.) and then drop the SI-units, then you'll be consistent. Then you're in pure mathematics and everythings is fabulously correct!

• This is why I switched to pure mathematics! – user50229 Dec 31 '12 at 17:23