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Difference between revisions of "2022 AMC 12A Problems/Problem 11"

(Created page with "==Problem== What is the product of all real numbers <math>x</math> such that the distance on the number line between <math>\log_6x</math> and <math>\log_69</math> is twice th...")
 
 
(19 intermediate revisions by 6 users not shown)
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<math>\textbf{(A) } 10 \qquad \textbf{(B) } 18 \qquad \textbf{(C) } 25 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 81</math>
 
<math>\textbf{(A) } 10 \qquad \textbf{(B) } 18 \qquad \textbf{(C) } 25 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 81</math>
 +
 +
==Solution 1==
 +
Let <math>a = 2 \cdot |\log_6 10 - 1| = |\log_6 9 - \log_6 x|  =  |\log_6 \frac{9}{x}| </math>.
 +
 +
<math> \pm a = \log_6 \frac{9}{x} \implies 6^{\pm a} = b^{\pm 1} = \frac{9}{x}  \implies x = 9 \cdot b^{\pm 1} </math>
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<math> 9b^1 \cdot 9b^{-1} = \boxed{81}</math>.
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 +
~ oinava
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 +
==Solution 2==
 +
First, notice that there must be two such numbers: one greater than <math>\log_69</math> and one less than it. Furthermore, they both have to be the same distance away, namely <math>2(\log_610 - 1)</math>. Let these two numbers be <math>\log_6a</math> and <math>\log_6b</math>. Because they are equidistant from <math>\log_69</math>, we have <math>\frac{\log_6a + \log_6b}{2} = \log_69</math>. Using log properties, this simplifies to <math>\log_6{\sqrt{ab}} = \log_69</math>. We then have <math>\sqrt{ab} = 9</math>, so <math>ab = \boxed{\textbf{(E) } 81}</math>.
 +
 +
~ jamesl123456
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 +
==Solution 3 (Logarithmic Rules and Casework)==
 +
 +
In effect we must find all <math>x</math> such that <math>\left|\log_6 9 - \log_6 x\right| = 2d</math> where <math>d = \log_6 10 - 1</math>.
 +
 +
Notice that by log rules
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<cmath>
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d = \log_6 10 - 1 = \log_6 \frac{10}{6}
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</cmath>
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Using log rules again,
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<cmath>
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2d = 2\log_6 \frac{10}{6} = \log_6 \frac{25}{9}
 +
</cmath>
 +
 +
Now we proceed by casework for the distinct values of <math>x</math>.
 +
 +
===Case 1===
 +
<cmath>
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\log_6 9 - \log_6 x_1 = 2d
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</cmath>
 +
Subbing in for <math>2d</math> and using log rules,
 +
<cmath>
 +
\log_6 \frac{9}{x_1} = \log_6 \frac{25}{9}
 +
</cmath>
 +
From this we may conclude that
 +
<cmath>
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\frac{9}{x_1} = \frac{25}{9} \implies x_1 = \frac{81}{25}
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</cmath>
 +
 +
===Case 2===
 +
<cmath>
 +
\log_6 9 - \log_6 x_2 = -2d
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</cmath>
 +
Subbing in for <math>-2d</math> and using log rules,
 +
<cmath>
 +
\log_6 \frac{9}{x_2} = \log_6 \frac{9}{25}
 +
</cmath>
 +
From this we conclude that
 +
<cmath>
 +
\frac{9}{x_2} = \frac{9}{25} \implies x_2 = 25
 +
</cmath>
 +
 +
Finding the product of the distinct values,
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<math>x_1x_2 = \boxed{\textbf{(E) } 81}</math>
 +
 +
~Spektrum
 +
 +
==Video Solution 1 (Quick and Simple)==
 +
https://youtu.be/2sqyO4SlFfc
 +
 +
~Education, the Study of Everything
 +
 +
==Video Solution 1 (Understand the question first)==
 +
https://youtu.be/7yAh4MtJ8a8?si=CsVGUSiyUiT4nNI0&t=2076
 +
 +
~Math-X
 +
== See Also ==
 +
{{AMC12 box|year=2022|ab=A|num-b=10|num-a=12}}
 +
{{MAA Notice}}

Latest revision as of 15:29, 25 October 2023

Problem

What is the product of all real numbers $x$ such that the distance on the number line between $\log_6x$ and $\log_69$ is twice the distance on the number line between $\log_610$ and $1$?

$\textbf{(A) } 10 \qquad \textbf{(B) } 18 \qquad \textbf{(C) } 25 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 81$

Solution 1

Let $a = 2 \cdot |\log_6 10 - 1| = |\log_6 9 - \log_6 x| = |\log_6 \frac{9}{x}|$.

$\pm a = \log_6 \frac{9}{x} \implies 6^{\pm a} = b^{\pm 1} = \frac{9}{x} \implies x = 9 \cdot b^{\pm 1}$

$9b^1 \cdot 9b^{-1} = \boxed{81}$.

~ oinava

Solution 2

First, notice that there must be two such numbers: one greater than $\log_69$ and one less than it. Furthermore, they both have to be the same distance away, namely $2(\log_610 - 1)$. Let these two numbers be $\log_6a$ and $\log_6b$. Because they are equidistant from $\log_69$, we have $\frac{\log_6a + \log_6b}{2} = \log_69$. Using log properties, this simplifies to $\log_6{\sqrt{ab}} = \log_69$. We then have $\sqrt{ab} = 9$, so $ab = \boxed{\textbf{(E) } 81}$.

~ jamesl123456

Solution 3 (Logarithmic Rules and Casework)

In effect we must find all $x$ such that $\left|\log_6 9 - \log_6 x\right| = 2d$ where $d = \log_6 10 - 1$.

Notice that by log rules \[d = \log_6 10 - 1 = \log_6 \frac{10}{6}\] Using log rules again, \[2d = 2\log_6 \frac{10}{6} = \log_6 \frac{25}{9}\]

Now we proceed by casework for the distinct values of $x$.

Case 1

\[\log_6 9 - \log_6 x_1 = 2d\] Subbing in for $2d$ and using log rules, \[\log_6 \frac{9}{x_1} = \log_6 \frac{25}{9}\] From this we may conclude that \[\frac{9}{x_1} = \frac{25}{9} \implies x_1 = \frac{81}{25}\]

Case 2

\[\log_6 9 - \log_6 x_2 = -2d\] Subbing in for $-2d$ and using log rules, \[\log_6 \frac{9}{x_2} = \log_6 \frac{9}{25}\] From this we conclude that \[\frac{9}{x_2} = \frac{9}{25} \implies x_2 = 25\]

Finding the product of the distinct values, $x_1x_2 = \boxed{\textbf{(E) } 81}$

~Spektrum

Video Solution 1 (Quick and Simple)

https://youtu.be/2sqyO4SlFfc

~Education, the Study of Everything

Video Solution 1 (Understand the question first)

https://youtu.be/7yAh4MtJ8a8?si=CsVGUSiyUiT4nNI0&t=2076

~Math-X

See Also

2022 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 10 		Followed by
Problem 12
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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