2020 AIME II Problems/Problem 3

Problem

The value of $x$ that satisfies $\log_{2^x} 3^{20} = \log_{2^{x+3}} 3^{2020}$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

Solution

Let $\log _{2^x}3^{20}=\log _{2^{x+3}}3^{2020}=n$ . Based on the equation, we get $(2^x)^n=3^{20}$ and $(2^{x+3})^n=3^{2020}$ . Expanding the second equation, we get $8^n\cdot2^{xn}=3^{2020}$ . Substituting the first equation in, we get $8^n\cdot3^{20}=3^{2020}$ , so $8^n=3^{2000}$ . Taking the 100th root, we get $8^{\frac{n}{100}}=3^{20}$ . Therefore, $(2^{\frac{3}{100}})^n=3^{20}$ , so $n=\frac{3}{100}$ and the answer is $\boxed{103}$ . ~rayfish

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2020 AIME II Problems/Problem 3

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