AMC12 2004 A

AMC12 2004 A · Q16

AMC12 2004 A · Q16. It mainly tests Exponents & radicals, Logarithms (rare).

The set of all real numbers $x$ for which \[\log_{2004}(\log_{2003}(\log_{2002}(\log_{2001}{x})))\] is defined is $\{x\mid x > c\}$. What is the value of $c$?

所有使得 \[\log_{2004}(\log_{2003}(\log_{2002}(\log_{2001}{x})))\] 有定义的实数 $x$ 的集合是 $\{x\mid x > c\}$。$c$ 的值是多少？

(A) 0 0

(B) $2001^{2002}$ $2001^{2002}$

(D) $2003^{2004}$ $2003^{2004}$

(E) $2001^{2002^{2003}}$ $2001^{2002^{2003}}$

Answer

Correct choice: (B)

正确答案：（B）

Solution

For all real numbers $a,b,$ and $c$ such that $b>1,$ note that: 1. $\log_b a$ is defined if and only if $a>0.$ 2. $\log_b a>c$ if and only if $a>b^c.$ Therefore, we have \begin{align*} \log_{2004}(\log_{2003}(\log_{2002}(\log_{2001}{x}))) \text{ is defined} &\implies \log_{2003}(\log_{2002}(\log_{2001}{x}))>0 \\ &\implies \log_{2002}(\log_{2001}{x})>1 \\ &\implies \log_{2001}{x}>2002 \\ &\implies x>2001^{2002}, \end{align*} from which $c=\boxed{\textbf {(B) }2001^{2002}}.$

对所有实数 $a,b,$ 和 $c$ 且 $b>1$，注意： 1. $\log_b a$ 有定义当且仅当 $a>0$。 2. $\log_b a>c$ 当且仅当 $a>b^c$。因此有 \begin{align*} \log_{2004}(\log_{2003}(\log_{2002}(\log_{2001}{x}))) \text{ 有定义} &\implies \log_{2003}(\log_{2002}(\log_{2001}{x}))>0 \\ &\implies \log_{2002}(\log_{2001}{x})>1 \\ &\implies \log_{2001}{x}>2002 \\ &\implies x>2001^{2002}, \end{align*} 由此 $c=\boxed{\textbf {(B) }2001^{2002}}$。

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