2003 AMC 12A Problem 24

Expand: $$\begin{aligned} &\log_a\dfrac ab+\log_b\dfrac ba \\ &\quad {}=(1-\log_a b)+(1-\log_b a) \\ &\quad {}=2-\left(\log_a b+\log_b a\right). \end{aligned}$$ Let $c=\log_a b\gt0.$ Since $c+\dfrac1c\ge2$ by AM-GM, the expression is at most $0.$ Equality holds when $c=1,$ that is, when $a=b,$ so the largest value is $0.$ Thus, the correct answer is B.