# Difference between revisions of "2016 AMC 12A Problems/Problem 16"

(Created page with "==Problem 16== The graphs of <math>y=\log_3 x, y=\log_x 3, y=\log_\frac{1}{3} x,</math> and <math>y=\log_x \dfrac{1}{3}</math> are plotted on the same set of axes. How many p...") |
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==Solution== | ==Solution== | ||

− | Setting the first two equations equal to each other, <math>\log_3 x = log_x 3</math>. | + | Setting the first two equations equal to each other, <math>\log_3 x = \log_x 3</math>. |

Solving this, we get <math>(3, 1)</math> and <math>(\frac{1}{3}, 1)</math>. | Solving this, we get <math>(3, 1)</math> and <math>(\frac{1}{3}, 1)</math>. |

## Revision as of 18:28, 4 February 2016

## Problem 16

The graphs of and are plotted on the same set of axes. How many points in the plane with positive -coordinates lie on two or more of the graphs?

## Solution

Setting the first two equations equal to each other, .

Solving this, we get and .

Similarly with the last two equations, we get and .

Now, by setting the first and third equations equal to each other, we get .

Pairing the first and fourth or second and third equations won't work because then .

Pairing the second and fourth equations will yield , but since you can't divide by , it doesn't work.

After trying all pairs, we have a total of solutions