2003 AMC 12A Problems/Problem 23

Revision as of 17:33, 22 February 2010 by Fuzzy growl (talk | contribs) (Solution)

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 $2-2=\boxed{0}.$