Does the logarithm of matrices with determinant -1 exist?

Question

I found on the book that $log(A)$ (where log(A) = $\sum_{i=1}^\infty (-1)^{i-1}(A-I)^i/i$) does not exist if $det(A)=-1$, is there any proof for that?

A useful information is that $log(A)$ converges if every entry of $A-I$ smaller than $1/n$ in magnitude.