Difference between revisions of "2004 AMC 12A Problems/Problem 16"
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~MRENTHUSIASM (Reconstruction) | ~MRENTHUSIASM (Reconstruction) | ||
| − | Another solution | + | Another solution (I did not know the proper methodology for doing this) |
2001^a=x | 2001^a=x | ||
Revision as of 21:44, 21 January 2023
Problem
The set of all real numbers 
for which
![\[\log_{2004}(\log_{2003}(\log_{2002}(\log_{2001}{x})))\]](http://latex.artofproblemsolving.com/e/f/a/efaf7382f309efe8361302a7c57315c0e92e4adb.png)
is defined is 
. What is the value of 
?
Solution
For all real numbers 
and such that 
note that:
is defined if and only if 
if and only if 
is defined if and only if 
if and only if 
Therefore, we have
from which 
~Azjps (Fundamental Logic)
~MRENTHUSIASM (Reconstruction)
Another solution (I did not know the proper methodology for doing this)
2001^a=x
2002^b = a
2003^c = b
2004^d = c
we can now rewrite the expression as x = 2001^(2002^(2003^(2004^d)))
the smallest value of x occurs when d approaches negative infinity. This makes the expression become
1.2001^(2002^(2003^0)) 2.2001^(2002^1)
which gives a final expression of 2001^2002
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 | |
