I am currently learning statistical physics and the logarithm of operators plays an important role, since the entropy is $S=k_B\ln\langle\rho\rangle$, where $\rho$ is a density operator.
Unfortunately, phisicists don't mention how $\ln\rho$ is defined and i couldn't find more information on the internet.
Since the exponential of an operator $A$ can be written as
$$e^A=\sum_{n=0}^{\infty}\frac{A^n}{n!}$$ and for $1-1<x\leq 1+1$ we have the formula $$\ln x=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}(x-1)^n}{n},$$ I was wondering whether I can calculate the logarithm of an operator by exchaning $x-1$ with $A-\mathrm{id}$, where id is the identity.