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Difference between revisions of "2020 AMC 12A Problems/Problem 10"

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<math>\textbf{(A) } 4 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 11 \qquad \textbf{(E) } 13</math>
 
<math>\textbf{(A) } 4 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 11 \qquad \textbf{(E) } 13</math>
  
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==Solution==
+
==Solution 1 (Properties of Logarithms)==
  
−
Any logarithm in the form <math>\log_{a^b} c = \frac{1}{b} \log_a c</math>. (this can be proved easily by using change of base formula to base <math>a</math>).
+
Any logarithm in the form <math>\log_{a^b} c = \frac{1}{b} \log_a c.</math> This can be proved easily by using change of base formula to base <math>a.</math>
  
−
so <cmath>\log_2{(\log_{2^4}{n})} = \log_{2^2}{(\log_{2^2}{n})}</cmath>
+
So, the original equation <math>\log_2{(\log_{2^4}{n})} = \log_{2^2}{(\log_{2^2}{n})}</math> becomes <cmath>\log_2\left({\frac{1}{4}\log_{2}{n}}\right) = \frac{1}{2}\log_2\left({\frac{1}{2}\log_2{n}}\right).</cmath>
 +
Using log property of addition, we expand both sides and then simplify:
 +
<cmath>\begin{align*}
 +
\log_2{\frac{1}{4}}+\log_2{(\log_{2}{n}}) &= \frac{1}{2}\left[\log_2{\frac{1}{2}} +\log_{2}{(\log_2{n})}\right] \\
 +
\log_2{\frac{1}{4}}+\log_2{(\log_{2}{n}}) &= \frac{1}{2}\left[-1 +\log_{2}{(\log_2{n})}\right] \\
 +
-2+\log_2{(\log_{2}{n}}) &= -\frac{1}{2}+ \frac{1}{2}(\log_{2}{(\log_2{n})}).
 +
\end{align*}</cmath>
 +
Subtracting <math>\frac{1}{2}(\log_{2}{(\log_2{n})})</math> from both sides and adding <math>2</math> to both sides gives us <cmath>\frac{1}{2}(\log_{2}{(\log_2{n})}) = \frac{3}{2}.</cmath>
 +
Multiplying by <math>2,</math> exponentiating, and simplifying gives us
 +
<cmath>\begin{align*}
 +
\log_{2}{(\log_2{n})} &= 3 \\
 +
\log_2{n}&=8 \\
 +
n&=256.
 +
\end{align*}</cmath>
 +
Adding the digits together, we have <math>2+5+6=\boxed{\textbf{(E) } 13}.</math>  
  
−
becomes
+
~quacker88 (Solution)
  
−
<cmath>\log_2({\frac{1}{4}\log_{2}{n}}) = \frac{1}{2}\log_2({\frac{1}{2}\log_2{n}})</cmath>
+
~MRENTHUSIASM (Reformatting)
  
−
Using <math>\log</math> property of addition, we can expand the parentheses into
+
==Solution 2 (Properties of Logarithms)==
 +
We will apply the following logarithmic identity:
 +
<cmath>\log_{p^k}{q^k}=\log_{p}{q},</cmath>
 +
which can be proven by the Change of Base Formula: <cmath>\log_{p^k}{q^k}=\frac{\log_{p}{q^k}}{\log_{p}{p^k}}=\frac{k\log_{p}{q}}{k}=\log_{p}{q}.</cmath>
 +
Note that <math>\log_{16}{n}\neq0,</math> so we rewrite the original equation as follows:
 +
<cmath>\begin{align*}
 +
\log_4{(\log_{16}{n})^2}&=\log_4{(\log_4{n})} \\
 +
(\log_{16}{n})^2&=\log_4{n} \\
 +
(\log_{16}{n})^2&=\log_{16}{n^2} \\
 +
(\log_{16}{n})^2&=2\log_{16}{n} \\
 +
\log_{16}{n}&=2,
 +
\end{align*}</cmath>
 +
from which <math>n=16^2=256.</math> The sum of its digits is <math>2+5+6=\boxed{\textbf{(E) } 13}.</math>
  
−
<cmath>\log_2{(\frac{1}{4})}+\log_2{(\log_{2}{n}}) = \frac{1}{2}(\log_2{(\frac{1}{2})} +\log_{2}{(\log_2{n})})</cmath>
+
~MRENTHUSIASM
  
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Expanding the RHS and simplifying the logs without variables, we have
+
==Solution 3 (Properties of Logarithms)==
−
 
+
Using the change of base formula on the RHS of the initial equation yields
−
<cmath>-2+\log_2{(\log_{2}{n}}) = -\frac{1}{2}+ \frac{1}{2}(\log_{2}{(\log_2{n})})</cmath>
+
<cmath> \log_2{(\log_{16}{n})} = \frac{\log_2{(\log_4{n})}}{\log_2{4}}. </cmath>
−
 
+
This means we can multiply each side by <math>2</math> for
−
Subtracting <math>\frac{1}{2}(\log_{2}{(\log_2{n})})</math> from both sides and adding <math>2</math> to both sides gives us
+
<cmath> \log_2{(\log_{16}{n})^2} = \log_2{(\log_4{n})}. </cmath>
−
 
+
Canceling out the logs gives
−
<cmath>\frac{1}{2}(\log_{2}{(\log_2{n})}) = \frac{3}{2}</cmath>
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<cmath> (\log_{16}{n})^2=\log_4{n}. </cmath>
−
 
+
We use change of base on the RHS to see that
−
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
+
<cmath>\begin{align*}
−
 
+
(\log_{16}{n})^2&=\frac{ \log_{16}{n}}{\log_{16}{4}} \\
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<cmath>(\log_{2}{(\log_2{n})}) = 3</cmath>
+
(\log_{16}{n})^2&=2 \log_{16}{n}.
−
 
+
\end{align*}</cmath>
−
<cmath>2^{\log_{2}{(\log_2{n})}} = 2^3</cmath>
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Substituting in <math> m = \log_{16}{n} </math> gives <math> m^2=2m, </math> so <math> m </math> is either <math>0</math> or <math>2.</math> Since <math> m=0 </math> yields no solution for <math>n</math> (since this would lead to use taking the log of <math>0</math>), we get <math> 2 = \log_{16}{n}, </math> or <math> n=16^2=256, </math> for the digit-sum of <math>2 + 5 + 6 = \boxed{\textbf{(E) } 13}.</math>
−
 
 
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<cmath>\log_2{n}=8</cmath>
 
−
 
 
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<cmath>2^{\log_2{n}}=2^8</cmath>
 
−
 
 
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<cmath>n=256</cmath>
 
  
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Adding the digits together, we have <math>2+5+6=\boxed{\textbf{(E) }13}</math> ~quacker88
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~aop2014
  
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==Solution 2==
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==Solution 4 (Exponential Form)==
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We know that, as the answer is an integer, <math>n</math> must be some power of <math>16</math>. Testing <math>16</math> yields
+
Suppose <math>\log_2(\log_{16}n)=k\implies\log_{16}n=2^k\implies n=16^{2^k}.</math> Similarly, we have <math>\log_4(\log_4 n)=k\implies \log_4 n=4^k\implies n=4^{4^k}.</math> Thus, we have <cmath>16^{2^k}=(4^2)^{2^k}=4^{2^{k+1}}</cmath> and <cmath>4^{4^k}=4^{2^{2k}},</cmath> so <math>k+1=2k\implies k=1.</math> Plugging this in to either one of the expressions for <math>n</math> gives <math>256</math>, and the requested answer is <math>2+5+6=\boxed{\textbf{(E) } 13}.</math>
−
<cmath> \log_2{(\log_{16}{16})} = \log_4{(\log_4{16})} </cmath>
 
−
<cmath> \log_2{1} = \log_4{2}</cmath>
 
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<cmath> 0 = \frac{1}{2}</cmath>
 
−
which does not work. We then try <math>256</math>, giving us
 
  
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<cmath> \log_2{(\log_{16}{256})} = \log_4{(\log_4{256})} </cmath>
+
==Solution 5 (Guess and Check)==
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<cmath> \log_2{2} = \log_4{4}</cmath>
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We know that, as the answer is an integer, <math>n</math> must be some power of <math>16.</math> Testing <math>16</math> yields
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<cmath> 1 = 1 </cmath>
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<cmath>\begin{align*}
−
which holds true. Thus, <math>n = 256</math>, so the answer is <math>2 + 5 + 6 = \boxed{\textbf{(E) }13}</math>.
+
\log_2{(\log_{16}{16})} &= \log_4{(\log_4{16})} \\
 +
\log_2{1} &= \log_4{2} \\
 +
0 &= \frac{1}{2},
 +
\end{align*}</cmath>
 +
which does not work. We then try <math>256,</math> giving us
 +
<cmath>\begin{align*}
 +
\log_2{(\log_{16}{256})} &= \log_4{(\log_4{256})} \\
 +
\log_2{2} &= \log_4{4} \\
 +
1 &= 1,
 +
\end{align*}</cmath>
 +
which holds true. Thus, <math>n = 256,</math> so the answer is <math>2 + 5 + 6 = \boxed{\textbf{(E) } 13}.</math>
  
 
(Don't use this technique unless you absolutely need to! Guess and check methods aren't helpful for learning math.)
 
(Don't use this technique unless you absolutely need to! Guess and check methods aren't helpful for learning math.)
  
−
~ciceronii
+
~ciceronii (Solution)
−
 
 
  
−
==Solution 3-Change of Base==
+
~MRENTHUSIASM (Reformatting)
−
Using the change of base formula on the RHS of the initial equation yields
 
−
<cmath> \log_2{(\log_{16}{n})} = \frac{\log_2{(\log_4{n})}}{\log_2{4}} </cmath>
 
−
This means we can multiply each side by 2 for
 
−
<cmath> \log_2{(\log_{16}{n})^2} = \log_2{(\log_4{n})} </cmath>
 
−
Canceling out the logs gives
 
−
<cmath> (\log_{16}{n})^2=\log_4{n} </cmath>
 
−
We use change of base on the RHS to see that <cmath> (log_{16}{n})^2=\frac{ log_{16}{n}}{\log_{16}{4}} </cmath> or <cmath> (log_{16}{n})^2= 2 log_{16}{n}</cmath> Substituting in <math> m = log_{16}{n} </math> gives <math> m^2=2m </math>, so <math> m </math> is either <math>0</math> or <math>2</math>. Since <math> m=0 </math> yields no solution for <math>n</math>(since a log cannot be equal to <math>0</math>), we get <math> 2 = log_{16}{n} </math>, or <math> n=16^2=256 </math>, for a sum of <math>2 + 5 + 6 = \boxed{\textbf{(E) }13}</math>. ~aop2014
 
  
−
==Video Solution==
+
==Video Solution 1==
 
https://youtu.be/fzZzGqNqW6U
 
https://youtu.be/fzZzGqNqW6U
  
 
~IceMatrix
 
~IceMatrix
 +
 +
== Video Solution 2==
 +
https://youtu.be/RdIIEhsbZKw?t=814
 +
 +
~ pi_is_3.14
  
 
==See Also==
 
==See Also==

Revision as of 06:02, 3 January 2022

Problem

There is a unique positive integer $n$ such that\[\log_2{(\log_{16}{n})} = \log_4{(\log_4{n})}.\]What is the sum of the digits of $n?$

$\textbf{(A) } 4 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 11 \qquad \textbf{(E) } 13$

Solution 1 (Properties of Logarithms)

Any logarithm in the form $\log_{a^b} c = \frac{1}{b} \log_a c.$ This can be proved easily by using change of base formula to base $a.$

So, the original equation $\log_2{(\log_{2^4}{n})} = \log_{2^2}{(\log_{2^2}{n})}$ becomes \[\log_2\left({\frac{1}{4}\log_{2}{n}}\right) = \frac{1}{2}\log_2\left({\frac{1}{2}\log_2{n}}\right).\] Using log property of addition, we expand both sides and then simplify: \begin{align*} \log_2{\frac{1}{4}}+\log_2{(\log_{2}{n}}) &= \frac{1}{2}\left[\log_2{\frac{1}{2}} +\log_{2}{(\log_2{n})}\right] \\ \log_2{\frac{1}{4}}+\log_2{(\log_{2}{n}}) &= \frac{1}{2}\left[-1 +\log_{2}{(\log_2{n})}\right] \\ -2+\log_2{(\log_{2}{n}}) &= -\frac{1}{2}+ \frac{1}{2}(\log_{2}{(\log_2{n})}). \end{align*} Subtracting $\frac{1}{2}(\log_{2}{(\log_2{n})})$ from both sides and adding $2$ to both sides gives us \[\frac{1}{2}(\log_{2}{(\log_2{n})}) = \frac{3}{2}.\] Multiplying by $2,$ exponentiating, and simplifying gives us \begin{align*} \log_{2}{(\log_2{n})} &= 3 \\ \log_2{n}&=8 \\ n&=256. \end{align*} Adding the digits together, we have $2+5+6=\boxed{\textbf{(E) } 13}.$

~quacker88 (Solution)

~MRENTHUSIASM (Reformatting)

Solution 2 (Properties of Logarithms)

We will apply the following logarithmic identity: \[\log_{p^k}{q^k}=\log_{p}{q},\] which can be proven by the Change of Base Formula: \[\log_{p^k}{q^k}=\frac{\log_{p}{q^k}}{\log_{p}{p^k}}=\frac{k\log_{p}{q}}{k}=\log_{p}{q}.\] Note that $\log_{16}{n}\neq0,$ so we rewrite the original equation as follows: \begin{align*} \log_4{(\log_{16}{n})^2}&=\log_4{(\log_4{n})} \\ (\log_{16}{n})^2&=\log_4{n} \\ (\log_{16}{n})^2&=\log_{16}{n^2} \\ (\log_{16}{n})^2&=2\log_{16}{n} \\ \log_{16}{n}&=2, \end{align*} from which $n=16^2=256.$ The sum of its digits is $2+5+6=\boxed{\textbf{(E) } 13}.$

~MRENTHUSIASM

Solution 3 (Properties of Logarithms)

Using the change of base formula on the RHS of the initial equation yields \[\log_2{(\log_{16}{n})} = \frac{\log_2{(\log_4{n})}}{\log_2{4}}.\] This means we can multiply each side by $2$ for \[\log_2{(\log_{16}{n})^2} = \log_2{(\log_4{n})}.\] Canceling out the logs gives \[(\log_{16}{n})^2=\log_4{n}.\] We use change of base on the RHS to see that \begin{align*} (\log_{16}{n})^2&=\frac{ \log_{16}{n}}{\log_{16}{4}} \\ (\log_{16}{n})^2&=2 \log_{16}{n}. \end{align*} Substituting in $m = \log_{16}{n}$ gives $m^2=2m,$ so $m$ is either $0$ or $2.$ Since $m=0$ yields no solution for $n$ (since this would lead to use taking the log of $0$), we get $2 = \log_{16}{n},$ or $n=16^2=256,$ for the digit-sum of $2 + 5 + 6 = \boxed{\textbf{(E) } 13}.$

~aop2014

Solution 4 (Exponential Form)

Suppose $\log_2(\log_{16}n)=k\implies\log_{16}n=2^k\implies n=16^{2^k}.$ Similarly, we have $\log_4(\log_4 n)=k\implies \log_4 n=4^k\implies n=4^{4^k}.$ Thus, we have \[16^{2^k}=(4^2)^{2^k}=4^{2^{k+1}}\] and \[4^{4^k}=4^{2^{2k}},\] so $k+1=2k\implies k=1.$ Plugging this in to either one of the expressions for $n$ gives $256$, and the requested answer is $2+5+6=\boxed{\textbf{(E) } 13}.$

Solution 5 (Guess and Check)

We know that, as the answer is an integer, $n$ must be some power of $16.$ Testing $16$ yields \begin{align*} \log_2{(\log_{16}{16})} &= \log_4{(\log_4{16})} \\ \log_2{1} &= \log_4{2} \\ 0 &= \frac{1}{2}, \end{align*} which does not work. We then try $256,$ giving us \begin{align*} \log_2{(\log_{16}{256})} &= \log_4{(\log_4{256})} \\ \log_2{2} &= \log_4{4} \\ 1 &= 1, \end{align*} which holds true. Thus, $n = 256,$ so the answer is $2 + 5 + 6 = \boxed{\textbf{(E) } 13}.$

(Don't use this technique unless you absolutely need to! Guess and check methods aren't helpful for learning math.)

~ciceronii (Solution)

~MRENTHUSIASM (Reformatting)

Video Solution 1

https://youtu.be/fzZzGqNqW6U

~IceMatrix

Video Solution 2

https://youtu.be/RdIIEhsbZKw?t=814

~ pi_is_3.14

See Also

2020 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 9 		Followed by
Problem 11
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

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