2022 AMC 12A Problem 11

Below is the professionally curated solution for Problem 11 of the 2022 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2022 AMC 12A solutions, or check the answer key.

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Concepts:logarithm absolute value

Difficulty rating: 1530

11.

What is the product of all real numbers $x$ such that the distance on the number line between $\log_6 x$ and $\log_6 9$ is twice the distance on the number line between $\log_6 10$ and $1?$

$10$

$18$

$25$

$36$

$81$

Solution:

The right-hand distance is $|\log_6 10-1|=\log_6\dfrac53,$ so twice it is $2\log_6\dfrac53=\log_6\dfrac{25}{9}.$

Thus $\left|\log_6\dfrac{x}{9}\right|=\log_6\dfrac{25}{9},$ giving $\dfrac{x}{9}=\dfrac{25}{9}$ or $\dfrac{x}{9}=\dfrac{9}{25},$ so $x=25$ or $x=\dfrac{81}{25}.$

Their product is $25\cdot\dfrac{81}{25}=81.$

Thus, the correct answer is E.

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