2024 AIME I Problems/Problem 2
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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&=10 \\
4y\log_yx&=10. \\
\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 (if you're bad at logs)
Convert the two equations into exponents:
Take the top equation to the power of .
.
\end{align*}
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 |
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