Difference between revisions of "2003 AMC 12A Problems/Problem 23"

Revision as of 13:55, 28 February 2010

2003 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 22	Followed by Problem 24
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All AMC 12 Problems and Solutions

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-== Problem ==
-If <math>a\geq b > 1,</math> what is the largest possible value of <math>\log_{a}(a/b) + \log_{b}(b/a)?</math>
-<math>
-\mathrm{(A)}\ -2      \qquad
-\mathrm{(B)}\ 0     \qquad
-\mathrm{(C)}\ 2      \qquad
-\mathrm{(D)}\ 3      \qquad
-\mathrm{(E)}\ 4
-</math>
-== 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>
-<cmath>=2-(\log_{a}b+\frac {1}{\log_{a}b})</cmath>
-Since <math>a</math> and <math>b</math> are both positive, using [[AM-GM]] gives that the term in parentheses must be at least <math>2</math>, so the largest possible values is <math>2-2=\boxed{0}.</math>
 == See Also ==