AMC12 2022 A

AMC12 2022 A · Q11

AMC12 2022 A · Q11. It mainly tests Logarithms (rare), Algebra misc.

What is the product of all real numbers $x$ such that the distance on the number line between $\log_6x$ and $\log_69$ is twice the distance on the number line between $\log_610$ and $1$?

所有实数 $x$ 的积，使得数轴上 $\log_6x$ 与 $\log_69$ 之间的距离是数轴上 $\log_610$ 与 $1$ 之间距离的两倍，是多少？

(A) 10 10

(B) 18 18

(D) 36 36

(E) 81 81

Answer

Correct choice: (E)

正确答案：（E）

Solution

Let $a = 2 \cdot |\log_6 10 - 1| = |\log_6 9 - \log_6 x| = \left|\log_6 \frac{9}{x}\right|$. $\pm a = \log_6 \frac{9}{x} \implies 6^{\pm a} = b^{\pm 1} = \frac{9}{x} \implies x = 9 \cdot b^{\pm 1}$ $9b^1 \cdot 9b^{-1} = \boxed{81}$.

令 $a = 2 \cdot |\log_6 10 - 1| = |\log_6 9 - \log_6 x| = \left|\log_6 \frac{9}{x}\right|$。 $\pm a = \log_6 \frac{9}{x} \implies 6^{\pm a} = 9^{\pm 1} = \frac{9}{x} \implies x = 9 \cdot 9^{\mp 1}$ $9\cdot9^1 \cdot 9\cdot9^{-1} = \boxed{81}$。

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