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

Easiest Solution

Recall the identity $\log_{a^n} b^{m} = \frac{m}{n}\log_{a} b$ (which is easily proven using exponents) Then this problem turns into $\frac{30}{x}\log_{2} 3 = \frac{3030}{x+3}\log_{2} 3$ Divide $\log_{2} 3$ from both sides. And we are left with $\frac{30}{x}=\frac{3030}{x+3}$ .Solving this simple equation we get $x = \tfrac{3}{100} \Rightarrow \boxed{103}$ ~mlgjeffdoge21

Solution 3 (Official MAA)

Using the Change of Base Formula to convert the logarithms in the given equation to base $2$ yields $\frac{\log_2 3^{20}}{\log_2 2^x} = \frac{\log_2 3^{2020}}{\log_2 2^{x+3}}, \text{~ and then ~} \frac{20\log_2 3}{x\cdot\log_2 2} = \frac{2020\log_2 3}{(x+3)\log_2 2}.$ Canceling the logarithm factors then yields $\frac{20}x = \frac{2020}{x+3},$ which has solution $x = \frac3{100}.$ The requested sum is $3 + 100 = 103$ .

Video Solution

https://youtu.be/lPr4fYEoXi0 ~ CNCM

Video Solution 2

https://www.youtube.com/watch?v=x0QznvXcwHY?t=528

~IceMatrix

2020 AIME II (Problems • Answer Key • Resources)
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