Difference between revisions of "2024 AIME I Problems/Problem 2"
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By properties of logarithms, we can simplify the given equation to <math>x\log_xy=4y\log_yx=10</math>. Let us break this into two separate equations: | By properties of logarithms, we can simplify the given equation to <math>x\log_xy=4y\log_yx=10</math>. Let us break this into two separate equations: | ||
\begin{align*} | \begin{align*} | ||
− | x\log_xy&= | + | x\log_xy&=14 \\ |
− | 4y\log_yx&= | + | 4y\log_yx&=14. \\ |
\end{align*} | \end{align*} | ||
We multiply the two equations to get: | We multiply the two equations to get: | ||
− | <cmath>4xy\left(\log_xy\log_yx\right)= | + | <cmath>4xy\left(\log_xy\log_yx\right)=196.</cmath> |
Also by properties of logarithms, we know that <math>\log_ab\cdot\log_ba=1</math>; thus, <math>\log_xy\cdot\log_yx=1</math>. Therefore, our equation simplifies to: | Also by properties of logarithms, we know that <math>\log_ab\cdot\log_ba=1</math>; thus, <math>\log_xy\cdot\log_yx=1</math>. Therefore, our equation simplifies to: | ||
− | <cmath>4xy= | + | <cmath>4xy=196\implies xy=\boxed{049}.</cmath> |
~Technodoggo | ~Technodoggo |
Revision as of 19:47, 2 February 2024
Contents
Problem
There exist real numbers and , both greater than 1, such that . Find .
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&=14 \\ 4y\log_yx&=14. \\ \end{align*} We multiply the two equations to get:
Also by properties of logarithms, we know that ; thus, . Therefore, our equation simplifies to:
~Technodoggo
Solution 2
Convert the two equations into exponents:
Take to the power of :
Plug this into :
So
~alexanderruan
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.