2022 AMC 12A Problem 11

Problem 11 of 25IntermediateAlgebra

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?

Show solution

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.

Tagged: logarithm · absolute value

More practice