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

2021 AMC 12A Spring Problem 14

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14.

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}

21

100\log_5 3

200\log_3 5

2{,}200

21{,}000

Use

\log_{a^k} b^m = \dfrac{m}{k}\log_a b

to simplify each term

The first sum is

\left(\sum k\right)\log_5 3

and the second is

100\log_3 5;

their logs are reciprocals

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.