2003 AMC 12A Problems/Problem 23

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 two, so the largest possible values is $2-2=\boxed{0}.$

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2003 AMC 12A Problems/Problem 23

Solution