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

Revision as of 15:43, 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 $a,b,$ and $c$ 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 $\begin{align*} \log_{2004}(\log_{2003}(\log_{2002}(\log_{2001}{x}))) \text{ is defined} &\implies \log_{2003}(\log_{2002}(\log_{2001}{x}))>0 \\ &\implies \log_{2002}(\log_{2001}{x})>1 \\ &\implies \log_{2001}{x}>2002 \\ &\implies x>2001^{2002}, \end{align*}$ from which $c=\boxed{\text {(B) }2001^{2002}}.$

~Azjps (Fundamental Logic)

~MRENTHUSIASM (Reconstruction)

2004 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 15	Followed by Problem 17
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

@@ Line 9: / Line 9: @@
 == Solution ==
-For all real numbers <math>a,b</math> and <math>c</math> such that <math>b>0</math> and <math>b\neq1,</math> note that:
+For all real numbers <math>a,b,</math> and <math>c</math> such that <math>b>0</math> and <math>b\neq1,</math> note that:
 <ol style="margin-left: 1.5em;">
    <li><math>\log_b a</math> is defined if and only if <math>a>0.</math></li><p>

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

Revision as of 15:43, 10 July 2021

Problem

Solution

See also