If $a\geq b > 1,$ what is the largest possible value of $\log_{a}(a/b) + \log_{b}(b/a)?$
$\mathrm{(A)}\ -2      \qquad \mathrm{(B)}\ 0     \qquad \mathrm{(C)}\ 2      \qquad \mathrm{(D)}\ 3      \qquad \mathrm{(E)}\ 4$
Using logarithmic rules, we see that
\[\log_{a}a-\log_{a}b+\log_{b}b-\log_{b}a = 2-(\log_{a}b+\log_{b}a)\]
\[=2-\left(\log_{a}b+\frac {1}{\log_{a}b}\right)\]
Since $a$ and $b$ are both greater than $1$, using [AM-GM](https://artofproblemsolving.com/wiki/index.php/AM-GM) gives that the term in parentheses must be at least $2$, so the largest possible values is $2-2=0 \Rightarrow \boxed{\textbf{B}}.$
Note that the maximum occurs when $a=b$.