Difference between revisions of "2024 AIME I Problems/Problem 2"
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− | + | 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>. | |
+ | |||
+ | ==Solution 1== | ||
+ | |||
+ | 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*} | ||
+ | x\log_xy&=10 \ | ||
+ | 4y\log_yx&=10. \ | ||
+ | \end{align*} | ||
+ | We multiply the two equations to get: | ||
+ | <cmath>4xy\left(\log_xy\log_yx\right)=100.</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: | ||
+ | |||
+ | <cmath>4xy=100\implies xy=\boxed{025}.</cmath> | ||
+ | |||
+ | ~Technodoggo |
Revision as of 12:42, 2 February 2024
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