2020 AMC 12A Problems/Problem 10

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 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 $\begin{align*} (\log_{2}{(\log_2{n})}) &= 3 \\ 2^{\log_{2}{(\log_2{n})}} &= 2^3 \\ \log_2{n}&=8 \\ 2^{\log_2{n}}&=2^8 \\ n&=256. \end{align*}$ Adding the digits together, we have $2+5+6=\boxed{\textbf{(E) }13}.$

~quacker88 (Solution)

~MRENTHUSIASM (Reformatting)

Solution 2

We 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 (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 $\begin{align*} (\log_{16}{n})^2&=\frac{ \log_{16}{n}}{\log_{16}{4}} \\ (\log_{16}{n})^2&=2 \log_{16}{n}. \end{align*}$ 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

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