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?