2003 AMC 12A Problems/Problem 23

Problem

If $a\geq b > 1,$ what is the largest possible value of $\log_{a}(a/b) + \log_{b}(b/a)?$

$\mathrm{(A)}\ -2 \qquad \mathrm{(B)}\ 0 \qquad \mathrm{(C)}\ 2 \qquad \mathrm{(D)}\ 3 \qquad \mathrm{(E)}\ 4$

Solution

Using logarithmic rules, we see that

$\log_{a}a-\log_{a}b+\log_{b}b-\log_{b}a = 2-(\log_{a}b+\log_{b}a$ $=2-(\log_{a}b+\frac {1}{\log_{a}b})$

Since $a$ and $b$ are both positive, using AM-GM gives that the term in parentheses must be at least $2$ , so the largest possible values is $2-2=\boxed{0}.$

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2003 AMC 12A Problems/Problem 23

Problem

Solution