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2024 AIME I Problems/Problem 2

Revision as of 11:50, 5 June 2024 by Grammaticus (talk | contribs) (→‎Solution 6)

Problem

There exist real numbers $x$ and $y$, both greater than 1, such that $\log_x\left(y^x\right)=\log_y\left(x^{4y}\right)=10$. Find $xy$.

Video Solution & More by MegaMath

https://www.youtube.com/watch?v=jxY7BBe-4gU

Solution 1

By properties of logarithms, we can simplify the given equation to $x\log_xy=4y\log_yx=10$. Let us break this into two separate equations:

\[x\log_xy=10\] \[4y\log_yx=10.\] We multiply the two equations to get: \[4xy\left(\log_xy\log_yx\right)=100.\]

Also by properties of logarithms, we know that $\log_ab\cdot\log_ba=1$; thus, $\log_xy\cdot\log_yx=1$. Therefore, our equation simplifies to:

\[4xy=100\implies xy=\boxed{025}.\]

~Technodoggo

Solution 2

Convert the two equations into exponents:

\[x^{10}=y^x~(1)\] \[y^{10}=x^{4y}~(2).\]

Take $(1)$ to the power of $\frac{1}{x}$:

\[x^{\frac{10}{x}}=y.\]

Plug this into $(2)$:

\[x^{(\frac{10}{x})(10)}=x^{4(x^{\frac{10}{x}})}\] \[{\frac{100}{x}}={4x^{\frac{10}{x}}}\] \[{\frac{25}{x}}={x^{\frac{10}{x}}}=y,\]

So $xy=\boxed{025}$

~alexanderruan

Solution 3

Similar to solution 2, we have:

$x^{10}=y^x$ and $y^{10}=x^{4y}$

Take the tenth root of the first equation to get

$x=y^{\frac{x}{10}}$

Substitute into the second equation to get

$y^{10}=y^{\frac{4xy}{10}}$

This means that $10=\frac{4xy}{10}$, or $100=4xy$, meaning that $xy=\boxed{25}$. ~MC413551


Solution 4

The same with other solutions, we have obtained $x^{10}=y^x$ and $y^{10}=x^{4y}$. Then, $x^{10}y^{10}=y^xx^{4y}$. So, $x^{10}=x^{4y}$, $y^{10}=y^{x}$. $x=10$ and $y=2.5$.

Solution 5

Using the first expression, we see that $x^{10} = y^x$. Now, taking the log of both sides, we get $\log_y(x^{10}) = \log_y(y^x)$. This simplifies to $10 \log_y(x) = x$. This is still equal to the second equation in the problem statement, so $10 \log_y(x) = x = 4y \log_y(x)$. Dividing by $\log_y(x)$ on both sides, we get $x = 4y = 10$. Therefore, $x = 10$ and $y = 2.5$, so $xy = \boxed{25}$.

~idk12345678


Solution 6

Put \[y=x^a\].We see: \[ax=10\] and \[4x^a/a=10\] which gives rise to \[xy=25\] which is the required answer.

~Grammaticus

Video Solution

https://youtu.be/qLUahGcewT4

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Video Solution

https://youtu.be/6C0yHp5GUBY

~Veer Mahajan

See also

2024 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 1 		Followed by
Problem 3
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All AIME Problems and Solutions

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