AMC12 2008 B

AMC12 2008 B · Q14

AMC12 2008 B · Q14. It mainly tests Logarithms (rare), Circle theorems.

A circle has a radius of $\log_{10}{(a^2)}$ and a circumference of $\log_{10}{(b^4)}$. What is $\log_{a}{b}$?

一个圆的半径为 $\log_{10}{(a^2)}$，周长为 $\log_{10}{(b^4)}$。求 $\log_{a}{b}$。

(A) $\frac{1}{4\pi}$ $\frac{1}{4\pi}$

(B) $\frac{1}{\pi}$ $\frac{1}{\pi}$

(D) $2\pi$ $2\pi$

(E) $4\pi$ $4\pi$

Answer

Correct choice: (C)

正确答案：（C）

Solution

Let $C$ be the circumference of the circle, and let $r$ be the radius of the circle. Using log properties, $C=\log_{10}{(b^4)}=4\log_{10}{(b)}$ and $r=\log_{10}{(a^2)}=2\log_{10}{(a)}$. Since $C=2\pi r$, $4\log_{10}{(b)}=2\pi\cdot2\log_{10}{(a)} \Rightarrow \log_{a}{b} = \frac{\log_{10}{(b)}}{\log_{10}{(a)}}=\pi \Rightarrow C$.

设 $C$ 为圆的周长，$r$ 为圆的半径。利用对数性质，$C=\log_{10}{(b^4)}=4\log_{10}{(b)}$ 且 $r=\log_{10}{(a^2)}=2\log_{10}{(a)}$。由于 $C=2\pi r$，有 $4\log_{10}{(b)}=2\pi\cdot2\log_{10}{(a)} \Rightarrow \log_{a}{b} = \frac{\log_{10}{(b)}}{\log_{10}{(a)}}=\pi \Rightarrow C$.

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