2024 AIME I Problems/Problem 2
Revision as of 13:42, 2 February 2024 by Technodoggo (talk | contribs)
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