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Is there any such thing as the logarithm of a unit? E.g. $log(kg)$?

I'm trying to prove the Buckingham-pi theorem through 3 lemmas.

Lemma 1: Assume $R_1, ..., R_n$ are given physical quantities and $r = rank(A)$. Then there are precisely $n - r$ independent dimensionless combinations $\pi_1, ..., \pi_{n-r}$.

I began by taking the logarithm on both sides of $1 = [\pi] = [R_1^{\lambda_1}, ..., R_n^{\lambda_n}]$, where $[R] = F_1^{a_1}, ..., F_l^{a_l}$, and the $F_i$'s are the units.

My problem is that I end up with taking the logarithm of a unit, e.g. $log(F_i)$. Is there any such thing as the logarithm of a unit?

harisf
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  • Aren't you just working with the exponents which add, and which are themselves dimensionless? – Mark Bennet Aug 21 '15 at 09:29
  • Yes, you're right. I managed to do without taking the logarithm and only considering the sum of the exponents. However, according to a proof suggestion, there is this approach where you use the logarithm on the units. I don't understand how that approach works. – harisf Aug 21 '15 at 10:15

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