2002 AMC 12P Problems/Problem 21

Problem

Let $a$ and $b$ be real numbers greater than $1$ for which there exists a positive real number $c,$ different from $1$ , such that

$2(\log_a{c} + \log_b{c}) = 9\log_{ab}{c}.$

Find the largest possible value of $\log_a b.$

$\text{(A) }\sqrt{2} \qquad \text{(B) }\sqrt{3} \qquad \text{(C) }2 \qquad \text{(D) }\sqrt{6} \qquad \text{(E) }3$

We may rewrite the given equation as $2(\frac {\log c}{\log a} + \frac {\log c}{\log b}) = \frac {9\log c}{\log a + \log b}$ . Since $c \neq 1$

2002 AMC 12P (Problems • Answer Key • Resources)
Preceded by Problem 20	Followed by Problem 22
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