Difference between revisions of "2004 AMC 12A Problems/Problem 16"

Revision as of 02:23, 10 July 2021

Problem

The set of all real numbers $x$ for which

$\log_{2004}(\log_{2003}(\log_{2002}(\log_{2001}{x})))$

is defined is $\{x|x > c\}$ . What is the value of $c$ ?

$\text {(A) } 0\qquad \text {(B) }2001^{2002} \qquad \text {(C) }2002^{2003} \qquad \text {(D) }2003^{2004} \qquad \text {(E) }2001^{2002^{2003}}$

Solution

For all real numbers $b$ such that $b>0$ and $b\neq1,$ note that:

$\log_b a$ is defined if and only if $a>0.$

$\log_b a>c$ if and only if $a>b^c.$

Therefore, we have $\begin{align*} \log_{2004}(\log_{2003}(\log_{2002}(\log_{2001}{x}))) \text{ is defined} &\implies \log_{2003}(\log_{2002}(\log_{2001}{x}))>0 \\ &\implies \log_{2002}(\log_{2001}{x})>1 \\ &\implies \log_{2001}{x}>2002 \\ &\implies x>\boxed{\text {(B) }2001^{2002}}. \end{align*}$ ~Azjps (Fundamental Logic)

~MRENTHUSIASM (Reconstruction)

2004 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 15	Followed by Problem 17
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Difference between revisions of "2004 AMC 12A Problems/Problem 16"

Revision as of 02:23, 10 July 2021

Problem

Solution

See also

@@ Line 19: / Line 19: @@
 &\implies \log_{2002}(\log_{2001}{x})>1 \\
 &\implies \log_{2001}{x}>2002 \\
-&\implies \boxed{\text {(B) }2001^{2002}}.
+&\implies x>\boxed{\text {(B) }2001^{2002}}.
 \end{align*}</cmath>
 ~Azjps (Fundamental Logic)