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2003 AMC 12A Problems/Problem 23: Difference between revisions

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== Problem 23 ==
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 ==
== Solution ==


Revision as of 17:34, 22 February 2010

Problem 23

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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