What is the value of (\sum_k=1^20 \log_5^k 3^k^2) × (\sum_k=1^100 \log_9^k 25^k)?

2021 AMC 12A Spring Problem 14 — Solution

2021 AMC 12A Spring Problem 14

Below is the professionally curated solution for Problem 14 of the 2021 AMC 12A Spring, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 12A Spring solutions, or check the answer key.

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Concepts:logarithm summation

Difficulty rating: 1780

14.

What is the value of $\left(\sum_{k=1}^{20} \log_{5^k} 3^{k^2}\right) \cdot \left(\sum_{k=1}^{100} \log_{9^k} 25^k\right)?$

$21$

$100\log_5 3$

$200\log_3 5$

$2{,}200$

$21{,}000$

Solution:

For the first sum, $\log_{5^k} 3^{k^2} = \dfrac{k^2}{k}\log_5 3 = k\log_5 3,$ so $\sum_{k=1}^{20} k\log_5 3 = \frac{20\cdot 21}{2}\log_5 3 = 210\log_5 3.$

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.

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