Difference between revisions of "2020 AMC 12A Problems/Problem 10"
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<cmath>\log_{2}{(\log_2{n})} = 3</cmath> | <cmath>\log_{2}{(\log_2{n})} = 3</cmath> | ||
| − | <cmath>\cancel{2^{\log_{2}}{(\log_2{n})}} = 2^3</cmath> | + | <cmath>\cancel{2^{\log_{2}}}{(\log_2{n})}} = 2^3</cmath> |
Revision as of 10:42, 1 February 2020
Problem
There is a unique positive integer 
such that![\[\log_2{(\log_{16}{n})} = \log_4{(\log_4{n})}.\]](http://latex.artofproblemsolving.com/d/2/3/d23cbff85ac447661855a4f0344f9c162ce14b2e.png)
What is the sum of the digits of 
Solution
Any logarithm in the form 
.
so ![\[\log_2{(\log_{2^4}{n})} = \log_{2^2}{(\log_{2^2}{n})}\]](http://latex.artofproblemsolving.com/6/b/c/6bcb5e9212f349f1435c4621656e0b9d6a378acd.png)
becomes
![\[\log_2({\frac{1}{4}\log_{2}{n}}) = \frac{1}{2}\log_2({\frac{1}{2}\log_2{n}})\]](http://latex.artofproblemsolving.com/7/a/a/7aa4e9f55b54bb45db9b85e3d507b33d2f5ab5d6.png)
Using 
property of addition, we can expand the parentheses into
![\[\log_2{(\frac{1}{4})}+\log_2{(\log_{2}{n}}) = \frac{1}{2}(\log_2{(\frac{1}{2})} +\log_{2}{(\log_2{n})})\]](http://latex.artofproblemsolving.com/d/4/2/d420413d862673d0086ee45c71a767445a4b4218.png)
Expanding the RHS and simplifying the logs without variables, we have
![\[-2+\log_2{(\log_{2}{n}}) = -\frac{1}{2}+ \frac{1}{2}(\log_{2}{(\log_2{n})})\]](http://latex.artofproblemsolving.com/b/0/4/b0493c284840502aa92807d9d246c61f0f2953e2.png)
Subtracting 
from both sides and adding 
to both sides gives us
![\[\frac{1}{2}(\log_{2}{(\log_2{n})}) = \frac{3}{2}\]](http://latex.artofproblemsolving.com/8/a/7/8a7e05fe9f9869dfc11c714d49a9c7e8528a2312.png)
Multiplying by , raising the logs to exponents of base to get rid of the logs and simplifying gives us
![\[\log_{2}{(\log_2{n})} = 3\]](http://latex.artofproblemsolving.com/6/b/e/6be272ed16ea3a4b91e00e986fe357b3fc81ddb3.png)
\[\cancel{2^{\log_{2}}}{(\log_2{n})}} = 2^3\] (Error compiling LaTeX. Unknown error_msg)
