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2022 AMC 12A Problem 11

Problem 11 of 25IntermediateAlgebra

What is the product of all real numbers xx such that the distance on the number line between log⁡6x\log_6 x and log⁡69\log_6 9 is twice the distance on the number line between log⁡610\log_6 10 and 1?1?

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Solution

The right-hand distance is ∣log⁡610−1∣=log⁡653,|\log_6 10-1|=\log_6\dfrac53, so twice it is 2log⁡653=log⁡6259.2\log_6\dfrac53=\log_6\dfrac{25}{9}. Thus ∣log⁡6x9∣=log⁡6259,\left|\log_6\dfrac{x}{9}\right|=\log_6\dfrac{25}{9}, giving x9=259\dfrac{x}{9}=\dfrac{25}{9} or x9=925,\dfrac{x}{9}=\dfrac{9}{25}, so x=25x=25 or x=8125.x=\dfrac{81}{25}. Their product is 25⋅8125=81.25\cdot\dfrac{81}{25}=81. Thus, the correct answer is E.

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Concepts: logarithm · absolute value

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.