Also, mathematically, there's nothing wrong as such with taking the logarithm of a value expressed in arbitrary units. The logarithm just turns unknown multiplicative factors into unknown additive terms. For example, $$\log(T\ {\rm K}) = \log(T) + \log(1\ {\rm K}).$$
Sure, you end up with a constant term of $\log(1\ {\rm K})$ which has no fixed value, but it's no worse in that regard than any other unknown constant. If it cancels out, great! If not, you just carry it around until you can do something to it (like, say, exponentiate it to get back a factor of $\rm K$ again).
As long as you agree that $T\ {\rm K}$ represents a positive scalar value, it has a well defined logarithm, even if its exact value is unknown. It's no less defined than $\log(2x) = \log(2) + \log(x)$, where $x$ is any unknown positive value. (Of course, you're free to work in a system where you arbitrarily insist that $1\ {\rm K}$ isn't a positive scalar and $\log(1\ {\rm K})$ isn't defined. But the point is that we can define it, in the "obvious" way, without introducing any inconsistencies.)
Ps. It's perhaps worth noting that the reason we can do that is because $\rm K$ is an ordinary multiplicative unit which obeys all the usual rules of arithmetic, such as $1\ {\rm K} + 1\ {\rm K} = 2\ {\rm K}$ and $1\ {\rm K} \cdot 1\ {\rm K} = 1\ {\rm K}^2$. We would not be able to do that for "affine units" like $\rm ^\circ C$, which already conceal an additive constant in the notation. Arguably, this is because writing $x\ {\rm ^∘C}=(273.15+x)\ {\rm K}$ is itself an abuse of notation, retained for historical reasons.