Difference between revisions of "2024 AIME I Problems/Problem 2"
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==Problem== | ==Problem== | ||
| − | There exist real numbers <math>x</math> and <math>y</math>, both greater than 1, such that <math>\log_x\left(y^x\right)=\log_y\left(x^{4y}\right)=10</math>. Find <math>xy</math>. | + | There exist real numbers <math>x</math> and <math>y</math>, both greater than 1, such that <math>\log_x\left(y^x\right)=\log_y\left(x^{4y}\right)=10</math>. Find <math>xy</math>. |
| + | |||
| + | ==Video Solution & More by MegaMath== | ||
| + | https://www.youtube.com/watch?v=jxY7BBe-4gU | ||
==Solution 1== | ==Solution 1== | ||
Revision as of 12:54, 9 February 2024
Contents
Problem
There exist real numbers 
and 
, both greater than 1, such that 
. Find 
.
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 
. Let us break this into two separate equations:
\begin{align*}
x\log_xy&=10 \\
4y\log_yx&=10. \\
\end{align*}
We multiply the two equations to get:
![\[4xy\left(\log_xy\log_yx\right)=100.\]](http://latex.artofproblemsolving.com/4/3/3/433f42f4fda44557d99bd60f5db220c4e0682a93.png)
Also by properties of logarithms, we know that 
; thus, 
. Therefore, our equation simplifies to:
![\[4xy=100\implies xy=\boxed{025}.\]](http://latex.artofproblemsolving.com/9/6/8/968ba9e7f231134a15dd1b22aca223a5d5d3947c.png)
~Technodoggo
Solution 2
Convert the two equations into exponents:
![\[x^{10}=y^x~(1)\]](http://latex.artofproblemsolving.com/b/f/2/bf2af4ec313d2cf6f52c0893f418be33bd47d8bd.png)
![\[y^{10}=x^{4y}~(2).\]](http://latex.artofproblemsolving.com/2/9/1/29142adda91dae7f81e9e988e12d5b0380e60fb4.png)
Take 
to the power of 
:
![\[x^{\frac{10}{x}}=y.\]](http://latex.artofproblemsolving.com/3/0/9/3095267bc5f7c2f59735b250a03277b63b994697.png)
Plug this into 
:
![\[x^{(\frac{10}{x})(10)}=x^{4(x^{\frac{10}{x}})}\]](http://latex.artofproblemsolving.com/1/d/6/1d6fb0629c4d87e36efa435c6628275423711151.png)
![\[{\frac{100}{x}}={4x^{\frac{10}{x}}}\]](http://latex.artofproblemsolving.com/a/2/5/a250ac7a3cdf8769d70835b33c4c0cbda19e19dc.png)
![\[{\frac{25}{x}}={x^{\frac{10}{x}}}=y,\]](http://latex.artofproblemsolving.com/5/6/3/563826c31e726fe406ecfa632e6635f3ee945c42.png)
So 
~alexanderruan
Solution 3
Similar to solution 2, we have:
and 
Take the tenth root of the first equation to get
Substitute into the second equation to get
This means that 
, or 
, meaning that 
.
~MC413551
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution
~Veer Mahajan
See also
| 2024 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 1 |
Followed by Problem 3 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
