Difference between revisions of "2021 AMC 12A Problems/Problem 14"

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==Video Solution by Punxsutawney Phil==
 
==Video Solution by Punxsutawney Phil==
 
https://youtube.com/watch?v=FD9BE7hpRvg&t=322s
 
https://youtube.com/watch?v=FD9BE7hpRvg&t=322s
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==Video Solution by Hawk Math==
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https://www.youtube.com/watch?v=AjQARBvdZ20
  
 
==See also==
 
==See also==
 
{{AMC12 box|year=2021|ab=A|num-b=13|num-a=15}}
 
{{AMC12 box|year=2021|ab=A|num-b=13|num-a=15}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 15:55, 11 February 2021

Problem

What is the value of\[\left(\sum_{k=1}^{20} \log_{5^k} 3^{k^2}\right)\cdot\left(\sum_{k=1}^{100} \log_{9^k} 25^k\right)?\]$\textbf{(A) }21 \qquad \textbf{(B) }100\log_5 3 \qquad \textbf{(C) }200\log_3 5 \qquad \textbf{(D) }2,200\qquad \textbf{(E) }21,000$

Solution

This equals \[\left(\sum_{k=1}^{20}k\log_5(3)\right)\left(\sum_{k=1}^{100}\log_9(25)\right)=\frac{20\cdot21}{2}\cdot\log_5(3)\cdot100\log_3(5)=\boxed{\textbf{(E)} 21000}\] ~JHawk0224

Video Solution by Punxsutawney Phil

https://youtube.com/watch?v=FD9BE7hpRvg&t=322s

Video Solution by Hawk Math

https://www.youtube.com/watch?v=AjQARBvdZ20

See also

2021 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 13
Followed by
Problem 15
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All AMC 12 Problems and Solutions

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