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Difference between revisions of "2022 AMC 12B Problems/Problem 16"

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Suppose <math>x</math> and <math>y</math> are positive real numbers such that
 
Suppose <math>x</math> and <math>y</math> are positive real numbers such that
  
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<math>x^y=2^{64}</math> and <math>(\log_2{x})^{\log_2{y}}=2^{27}</math>.
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<math>x^y=2^{64}</math> and <math>(\log_2{x})^{\log_2{y}}=2^{7}</math>.
  
 
What is the greatest possible value of <math>\log_2{y}</math>?
 
What is the greatest possible value of <math>\log_2{y}</math>?

Revision as of 21:26, 20 November 2022

Problem

Suppose $x$ and $y$ are positive real numbers such that

$x^y=2^{64}$ and $(\log_2{x})^{\log_2{y}}=2^{7}$.

What is the greatest possible value of $\log_2{y}$?

$\textbf{(A) }3 \qquad \textbf{(B) }4 \qquad \textbf{(C) }3+\sqrt{2} \qquad \textbf{(D) }4+\sqrt{3} \qquad \textbf{(E) }7$

Solution

Take the base-two logarithm of both equations to get \[y\log_2 x = 64\quad\text{and}\quad (\log_2 y)(\log_2\log_2 x) = 7.\] Now taking the base-two logarithm of the first equation again yields \[\log_2 y + \log_2\log_2 x = 6.\] It follows that the real numbers $r:=\log_2 y$ and $s:=\log_2\log_2 x$ satisfy $r+s=6$ and $rs = 7$. Solving this system yields \[\{\log_2 y,\log_2\log_2 x\}\in\{3-\sqrt 2, 3 + \sqrt 2\}.\] Thus the largest possible value of $\log_2 y$ is $3+\sqrt 2 \implies \boxed{\textbf {(C)}}$.

cr. djmathman

Solution 2

$x^y=2^{64} \Rightarrow y\log_2{x}=64 \Rightarrow \log_2{x}=\dfrac{64}{y}$.

Substitution into $(\log_2{x})^{\log_2{y}}=2^{7}$ yields

$(\dfrac{64}{y})^{\log_2{y}}=2^{7} \Rightarrow \log_2{y}\log_2{\dfrac{64}{y}}=7 \Rightarrow \log_2{y}(6-\log_2{y})=7 \Rightarrow \log^2_2{y}-6\log_2{y}+7=0$.

Solving for $\log_2{y}$ yields $\log_2{y}=3-\sqrt{2}$ or $3+\sqrt{2}$, and we take the greater value $\boxed{\boldsymbol{(\textbf{C})3+\sqrt{2}}}$.

~4SunnyH

See Also

2022 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 15 		Followed by
Problem 17
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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