2003 AMC 12A Problem 24

Problem 24 of 25HarderAlgebra

a \ge b \gt 1,

what is the largest possible value of

\log_a(\frac{a}{b}) + \log_b(\frac{b}{a})?

Show solution

Solution

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.