2007 AMC 12A Problem 23

Below is the professionally curated solution for Problem 23 of the 2007 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2007 AMC 12A solutions, or check the answer key.

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Concepts:logarithm coordinate geometry square (geometry)

Difficulty rating: 1990

23.

Square $ABCD$ has area $36,$ and $AB$ is parallel to the $x$ -axis. Vertices $A,$ $B,$ and $C$ are on the graphs of $y=\log_a x,$ $y=2\log_a x,$ and $y=3\log_a x,$ respectively. What is $a?$

$\sqrt[6]{3}$

$\sqrt3$

$\sqrt[3]{6}$

$\sqrt6$

$6$

Solution:

Let $A=(p,\log_a p)$ and $B=(q,2\log_a q).$ Since $AB$ is horizontal, $\log_a p=2\log_a q=\log_a q^2,$ so $p=q^2.$

The side length is $6=|p-q|=|q^2-q|,$ whose only positive solution is $q=3.$

Since $C=(q,3\log_a q),$ the vertical side gives $BC=6=\log_a q=\log_a 3.$ Thus $a^6=3,$ so $a=\sqrt[6]{3}.$

Thus, the correct answer is A.

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