2003 AMC 12A Problems/Problem 23
Revision as of 17:32, 22 February 2010 by Fuzzy growl (talk | contribs) (Created page with '== Solution == Using logarithmic rules, we see that <cmath>\log_{a}a-\log_{a}b+\log_{b}b-\log_{b}a = 2-(\log_{a}b+\log_{b}a</cmath> <math>=2-(\log_{a}b+\frac {1}{\log_{a}b}</ma…')
Solution
Using logarithmic rules, we see that
![\[\log_{a}a-\log_{a}b+\log_{b}b-\log_{b}a = 2-(\log_{a}b+\log_{b}a\]](http://latex.artofproblemsolving.com/9/e/9/9e95d3760b7b2da7f5478020c500dc7d4e724758.png)
a
b![$are both positive, using [[AM-GM]] gives that the term in parentheses must be at least two, so the largest possible values is$](http://latex.artofproblemsolving.com/d/a/5/da5393dbb9512e466e279fe02740b002fe8ef213.png)
2-2=\boxed{0}.$
