2024 AIME I Problems/Problem 2
Contents
[hide]Problem
There exist real numbers and , both greater than 1, such that . Find .
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Video Solution By MathTutorZhengFrSG
~MathTutorZhengFrSG
Solution 1
By properties of logarithms, we can simplify the given equation to . Let us break this into two separate equations:
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
Solution 3
Similar to solution 2, we have:
and
Take the tenth root of the first equation to get
Substitute into the second equation to get
This means that , or , meaning that .
~MC413551
Solution 4
The same with other solutions, we have obtained and . Then, . So, an obvious solution is to have and . Solving, we get and .
Solution 5
Using the first expression, we see that . Now, taking the log of both sides, we get . This simplifies to . This is still equal to the second equation in the problem statement, so . Dividing by on both sides, we get . Therefore, and , so .
~idk12345678
Solution 6
Put .We see: and which gives rise to which is the required answer.
~Grammaticus
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution
~Veer Mahajan
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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