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

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==Solution 2==
 
==Solution 2==
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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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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
−
<cmath> \log_2{(\log_{16}{16})} = \log_4{(\log_4{16})} </cmath>
+
<cmath>\begin{align*}
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<cmath> \log_2{1} = \log_4{2}</cmath>
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\log_2{(\log_{16}{16})} &= \log_4{(\log_4{16})} \\
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<cmath> 0 = \frac{1}{2}</cmath>
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\log_2{1} &= \log_4{2} \\
−
which does not work. We then try <math>256</math>, giving us
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0 &= \frac{1}{2},
−
 
+
\end{align*}</cmath>
−
<cmath> \log_2{(\log_{16}{256})} = \log_4{(\log_4{256})} </cmath>
+
which does not work. We then try <math>256,</math> giving us
−
<cmath> \log_2{2} = \log_4{4}</cmath>
+
<cmath>\begin{align*}
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<cmath> 1 = 1 </cmath>
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\log_2{(\log_{16}{256})} &= \log_4{(\log_4{256})} \\
−
which holds true. Thus, <math>n = 256</math>, so the answer is <math>2 + 5 + 6 = \boxed{\textbf{(E) }13}</math>.
+
\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.)

Revision as of 07:12, 15 August 2021

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

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 can expand the parentheses into \[\log_2{\frac{1}{4}}+\log_2{(\log_{2}{n}}) = \frac{1}{2}\left[\log_2{\frac{1}{2}} +\log_{2}{(\log_2{n})}\right].\] Expanding the RHS and simplifying the logs without variables, we have \[-2+\log_2{(\log_{2}{n}}) = -\frac{1}{2}+ \frac{1}{2}(\log_{2}{(\log_2{n})}).\] 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 \\ 2^{\log_{2}{(\log_2{n})}} &= 2^3 \\ \log_2{n}&=8 \\ 2^{\log_2{n}}&=2^8 \\ n&=256. \end{align*} Adding the digits together, we have $2+5+6=\boxed{\textbf{(E) }13}.$

~quacker88 (Solution)

~MRENTHUSIASM (Reformatting)

Solution 2

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 3 (Change of Base)

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 \[(\log_{16}{n})^2=\frac{ \log_{16}{n}}{\log_{16}{4}}\] or \[(\log_{16}{n})^2= 2 \log_{16}{n}\] 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 a sum of $2 + 5 + 6 = \boxed{\textbf{(E) }13}$. ~aop2014

Solution 4

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

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

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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