2020 AIME II Problems/Problem 3
The value of that satisfies can be written as , where and are relatively prime positive integers. Find .
Let . Based on the equation, we get and . Expanding the second equation, we get . Substituting the first equation in, we get , so . Taking the 100th root, we get . Therefore, , and using the our first equation(), we get and the answer is . ~rayfish
Recall the identity (which is easily proven using exponents or change of base). Then this problem turns into Divide from both sides. And we are left with .Solving this simple equation we get ~mlgjeffdoge21
Because we have that or Since and thus resulting in or We remove the base 3 logarithm and the power of 2 to yield or
Our answer is ~ OreoChocolate
Solution 3 (Official MAA)
Using the Change of Base Formula to convert the logarithms in the given equation to base yields Canceling the logarithm factors then yieldswhich has solution The requested sum is .
https://youtu.be/lPr4fYEoXi0 ~ CNCM
Video Solution 2
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Video Solution 5
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