2021 AMC 12B Problems/Problem 21

Problem

Let $S$ be the sum of all positive real numbers $x$ for which $x^{2^{\sqrt2}}=\sqrt2^{2^x}.$ Which of the following statements is true?

$\textbf{(A) }S<\sqrt2 \qquad \textbf{(B) }S=\sqrt2 \qquad \textbf{(C) }\sqrt2<S<2\qquad \textbf{(D) }2\le S<6 \qquad \textbf{(E) }S\ge 6$

Solution (Rough Approximation)

Note that this solution is not recommended unless you're running out of time.

Upon pure observation, it is obvious that one solution to this equality is $x=\sqrt{2}$ . From this, we can deduce that this equality has two solutions, since $\sqrt{2}^{2^{x}}$ grows faster than $x^{2^{\sqrt{2}}}$ (for greater values of $x$ ) and $\sqrt{2}^{2^{x}}$ is greater than $x^{2^{\sqrt{2}}}$ for $x<\sqrt{2}$ and less than $x^{2^{\sqrt{2}}}$ for $\sqrt{2}<x<n$ , where $n$ is the second solution. Thus, the answer cannot be $\text{A}$ or $\text{B}$ . We then start plugging in numbers to roughly approximate the answer. When $x=2$ , $x^{2^{\sqrt{2}}}>\sqrt{2}^{2^{x}}$ , thus the answer cannot be $\text{C}$ . Then, when $x=4$ , $x^{2^{\sqrt{2}}}=4^{2^{\sqrt{2}}}<64<\sqrt{2}^{2^{x}}=256$ . Therefore, $S<4+\sqrt{2}<6$ , so the answer is $\boxed{\textbf{(D) } 2 \le S < 6}$ . ~Baolan

Solution 2

$x^{2^{\sqrt{2}}} = {\sqrt{2}}^{2^x}$

$2^{\sqrt{2}} \log x = 2^{x} \log \sqrt{2}$ (At this point we see by inspection that $x=\sqrt{2}$ is a solution.)

$\sqrt{2} \log 2 + \log \log x = x \log 2 + \log \log \sqrt{2}$

$\sqrt{2} + \log_2 \log_2 x = x + \log_2 \log_2 \sqrt{2} = x -1.$

$\log_2 \log_2 x = x - 1 - \sqrt{2}.$

RHS is a line. LHS is a concave curve that looks like a logarithm and has $x$ intercept at $(2,0).$ There are at most two solutions, one of which is $\sqrt{2}.$ But note that at $x=2,$ we have $\log_2 \log_2 (2) = 0 > 2 - 1 - \sqrt{2},$ meaning that the log log curve is above the line, so it must intersect the line again at a point $x > 2.$ Now we check $x=4$ and see that $\log_2 \log_2 (4) = 1 < 4 - 1 - \sqrt{2},$ which means at $x=4$ the line is already above the log log curve. Thus, the second solution lies in the interval $(2,4).$ The answer is $\boxed{2 \leq S < 6}.$

~ ccx09

Solution 3 (Cleaner Solution by Graphing and Very Light Approximations)

We rewrite the right side, then take the base- $2$ logarithm for both sides: $\begin{align*} x^{2^{\sqrt2}}&=\left(2^\frac12\right)^{2^x} \\ x^{2^{\sqrt2}}&=2^{\frac12\cdot2^x} \\ x^{2^{\sqrt2}}&=2^{2^{x-1}} \\ \log_2{\left(x^{2^{\sqrt2}}\right)}&=\log_2{\left(2^{2^{x-1}}\right)} \\ 2^{\sqrt2}\log_2{x}&=2^{x-1}. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (*) \end{align*}$ By observations, $x=\sqrt2$ is one solution. By quick sketches of $f(x)=2^{\sqrt2}\log_2{x}$ and $g(x)=2^{x-1},$ we know that $(*)$ has two solutions, with $x=\sqrt2$ the smaller solution. We construct the following table of values: $\begin{array}{c|c|c|c} \boldsymbol{x} & \boldsymbol{f(x)=2^{\sqrt2}\log_2{x}} & \boldsymbol{g(x)=2^{x-1}} & \textbf{Comparison} \\ [1ex] \hline & & & \\ [-1ex] 1 & 0 & 1 & \\ [1ex] \sqrt2 & 2^{\sqrt2-1} & 2^{\sqrt2-1} & f\left(\sqrt2\right)=g\left(\sqrt2\right) \\ [1ex] 2 & 2^{\sqrt2} & 2 & f(2)>g(2) \\ [1ex] 3 & 2^{\sqrt2}\log_2{3} & 2^2 & \\ [1ex] 4 & 2^{\sqrt2+1} & 2^3 & f(4)<g(4) \end{array}$ Let $x=t$ be the larger solution. Since exponential functions outgrow logarithmic functions, we have $f(x)<g(x)$ for all $x>t.$ By the Intermediate Value Theorem, we get that $t\in(2,4).$ Finally, we conclude that $S=\sqrt2+t\in\left(\sqrt2+2,\sqrt2+4\right),$ and the answer must be $\boxed{\textbf{(D) }2\le S<6}.$

Graphs of $f(x)$ and $g(x)$ in Desmos: https://www.desmos.com/calculator/rjlgnikxaz

~MRENTHUSIASM

Video Solution by OmegaLearn (Logarithmic Tricks)

https://youtu.be/uCTpLB-kGR4

~ pi_is_3.14

Video Solution by hippopotamus1:

https://www.youtube.com/watch?v=GjO6C_qC13U&feature=youtu.be

2021 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 20	Followed by Problem 22
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