Difference between revisions of "2004 AMC 12A Problems/Problem 16"
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<cmath>\log_{2004}(\log_{2003}(\log_{2002}(\log_{2001}{x})))</cmath> | <cmath>\log_{2004}(\log_{2003}(\log_{2002}(\log_{2001}{x})))</cmath> | ||
− | is defined is <math>\{x | + | is defined is <math>\{x\mid x > c\}</math>. What is the value of <math>c</math>? |
− | <math>\ | + | <math>\textbf {(A) } 0\qquad \textbf {(B) }2001^{2002} \qquad \textbf {(C) }2002^{2003} \qquad \textbf {(D) }2003^{2004} \qquad \textbf {(E) }2001^{2002^{2003}}</math> |
− | == Solution == | + | == Solution 1 == |
− | For all real numbers <math>a,b,</math> and <math>c</math> such that <math>b> | + | For all real numbers <math>a,b,</math> and <math>c</math> such that <math>b>1,</math> note that: |
<ol style="margin-left: 1.5em;"> | <ol style="margin-left: 1.5em;"> | ||
<li><math>\log_b a</math> is defined if and only if <math>a>0.</math></li><p> | <li><math>\log_b a</math> is defined if and only if <math>a>0.</math></li><p> | ||
Line 21: | Line 21: | ||
&\implies x>2001^{2002}, | &\implies x>2001^{2002}, | ||
\end{align*}</cmath> | \end{align*}</cmath> | ||
− | from which <math>c=\boxed{\ | + | from which <math>c=\boxed{\textbf {(B) }2001^{2002}}.</math> |
− | ~Azjps | + | ~Azjps ~MRENTHUSIASM |
− | + | == Solution 2 == | |
+ | Let | ||
+ | <cmath>\begin{align*} | ||
+ | x &= 2001^a, \\ | ||
+ | a &= 2002^b, \\ | ||
+ | b &= 2003^c, \\ | ||
+ | c &= 2004^d. | ||
+ | \end{align*}</cmath> | ||
+ | It follows that <cmath>x = 2001^{2002^{2003^{2004^d}}}.</cmath> | ||
+ | The smallest value of <math>x</math> occurs when <math>d\rightarrow -\infty,</math> so this expression becomes | ||
+ | <cmath>x = 2001^{2002^{2003^0}} = 2001^{2002^1} = \boxed{\textbf {(B) }2001^{2002}}.</cmath> | ||
+ | |||
+ | ==Video Solution (Logical Thinking)== | ||
+ | https://youtu.be/46c-VN1QzWk | ||
+ | |||
+ | ~Education, the Study of Everything | ||
== See also == | == See also == |
Latest revision as of 02:01, 23 January 2023
Problem
The set of all real numbers for which
is defined is . What is the value of
?
Solution 1
For all real numbers and
such that
note that:
is defined if and only if
if and only if
Therefore, we have
from which
~Azjps ~MRENTHUSIASM
Solution 2
Let
It follows that
The smallest value of
occurs when
so this expression becomes
Video Solution (Logical Thinking)
~Education, the Study of Everything
See also
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 |