2021 AMC 12A Problem 14

Problem 14 of 25IntermediateAlgebra

What is the value of

\begin{aligned} &\left(\sum_{k=1}^{20} \log_{5^k} 3^{k^2}\right) \\ &\quad {}\cdot \left(\sum_{k=1}^{100} \log_{9^k} 25^k\right)? \end{aligned}

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Solution

For the first sum,

\log_{5^k} 3^{k^2} = \dfrac{k^2}{k}\log_5 3 = k\log_5 3,

\begin{aligned} &\sum_{k=1}^{20} k\log_5 3 = \frac{20\cdot 21}{2}\log_5 3 \\ &= 210\log_5 3. \end{aligned}

For the second sum,

\log_{9^k} 25^k = \log_9 25 = \log_3 5,

independent of

k,

so the sum is

100\log_3 5.

Since

\log_5 3 \cdot \log_3 5 = 1,

the product is

210 \cdot 100 = 21{,}000.

Thus, the correct answer is E.