2022 AMC 12A Problem 11

Attempt Problem 11 of the 2022 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 2022 AMC 12A solutions, or check the answer key.

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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

\log_6 10-1=\log_6\dfrac{10}{6}=\log_6\dfrac53

The condition is

\left|\log_6\dfrac{x}{9}\right|=\log_6\dfrac{25}{9}

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}

\dfrac{x}{9}=\dfrac{9}{25},

x=25

x=\dfrac{81}{25}.

Their product is

25\cdot\dfrac{81}{25}=81.

Thus, the correct answer is E.

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