Difference between revisions of "2020 AMC 12A Problems/Problem 10"
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Multiplying by <math>2</math>, raising the logs to exponents of base <math>2</math> to get rid of the logs and simplifying gives us | Multiplying by <math>2</math>, raising the logs to exponents of base <math>2</math> to get rid of the logs and simplifying gives us | ||
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+ | <cmath>(\log_{2}{(\log_2{n})}) = 3</cmath> | ||
<cmath>2^{\log_{2}{(\log_2{n})}} = 2^3</cmath> | <cmath>2^{\log_{2}{(\log_2{n})}} = 2^3</cmath> | ||
+ | |||
+ | <cmath>\log_2{n}=8</cmath> | ||
+ | |||
+ | <cmath>2^{\log_2{n}=2^8</cmath> | ||
+ | |||
+ | <math></math>n=256<math> | ||
+ | |||
+ | Adding the digits together, we have </math>2+5+6=\boxed{\textbf{E) }13}$ |
Revision as of 10:45, 1 February 2020
Problem
There is a unique positive integer such thatWhat is the sum of the digits of
Solution
Any logarithm in the form .
so
becomes
Using property of addition, we can expand the parentheses into
Expanding the RHS and simplifying the logs without variables, we have
Subtracting from both sides and adding to both sides gives us
Multiplying by , raising the logs to exponents of base to get rid of the logs and simplifying gives us
\[2^{\log_2{n}=2^8\] (Error compiling LaTeX. Unknown error_msg)
$$ (Error compiling LaTeX. Unknown error_msg)n=2562+5+6=\boxed{\textbf{E) }13}$