2007 AMC 12A Problem 23

Attempt Problem 23 of the 2007 AMC 12A below, then check your answer against the professionally curated solution 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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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

Since

AB

is horizontal with length

6,

set

\log_a p=2\log_a q,

giving

p=q^2

Solve

|q^2-q|=6

for

q,

then use

BC=6=\log_a q

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,

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,

a=\sqrt[6]{3}.

Thus, the correct answer is A.

Problem 23 in other years