If a ≥ b > 1, what is the largest possible value of log_a((a)/(b)) + log_b((b)/(a))?

2003 AMC 12A Problem 24

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24.

a \ge b \gt 1,

what is the largest possible value of

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

-2

0

2

3

4

Simplify to

2-\log_a b-\log_b a

With

c=\log_a b\gt0,

use

c+\dfrac1c\ge2

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.