Difference between revisions of "2024 AIME I Problems/Problem 2"
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~Technodoggo | ~Technodoggo | ||
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+ | ==Solution 2 (if you're bad at logs) | ||
+ | |||
+ | Convert the two equations into exponents. | ||
+ | \begin{align*} | ||
+ | <math>x^{10}=y^x</math> \ | ||
+ | <math>y^{10}=x^{4y}</math> \ | ||
+ | \end{align*} | ||
+ | |||
==See also== | ==See also== |
Revision as of 18:41, 2 February 2024
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:
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.
\begin{align*}
\
\
\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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.