2016 AMC 12A Problem 16

Question

The graphs of

y=\log_3 x,

y=\log_x 3,

y=\log_{\frac{1}{3}} x,

and

y=\log_x\dfrac{1}{3}

are plotted on the same set of axes. How many points in the plane with positive

x

-coordinates lie on two or more of the graphs?

2

3

4

5

6

Let

u=\log_3 x

and rewrite all four as

u,\ \dfrac1u,\ -u,\ -\dfrac1u

Find every

x

where two of these four expressions are equal

Let

u=\log_3 x.

Then

\log_x 3=\dfrac1u,

\log_{\frac{1}{3}}x=-u,

and

\log_x\dfrac13=-\dfrac1u.

Two graphs meet where two of

u,\dfrac1u,-u,-\dfrac1u

are equal for some valid

x\gt 0.

Setting

u=\dfrac1u

gives

u=\pm1,

so

x=3

or

x=\dfrac13;

setting

-u=-\dfrac1u

gives the same values. Setting

u=-u

gives

u=0,

i.e.

x=1,

where

\log_3 x

and

\log_{\frac{1}{3}}x

are both

0.

The remaining pairings have no real solution.

The distinct intersection points are

(1,0),

(3,1),

\left(\dfrac13,-1\right),

(3,-1),

and

\left(\dfrac13,1\right),

so there are

5.

Thus, the correct answer is D.

Problem 16 in other years