Difference between revisions of "2018 AMC 12B Problems/Problem 7"
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== Solution 1 == | == Solution 1 == | ||
| − | From the Change of Base Formula, we have <cmath>\frac{\prod_{i=3}^{13} \log (2i+1)}{\prod_{i=1}^{11}\log (2i+1)} = \frac{\log 25 \cdot \log 27}{\log 3 \cdot \log 5} = \frac{(2\log 5)\cdot(3\log 3}{\log 3 \cdot \log 5} = \boxed{\textbf{(C) } 6}.</cmath> | + | From the Change of Base Formula, we have <cmath>\frac{\prod_{i=3}^{13} \log (2i+1)}{\prod_{i=1}^{11}\log (2i+1)} = \frac{\log 25 \cdot \log 27}{\log 3 \cdot \log 5} = \frac{(2\log 5)\cdot(3\log 3)}{\log 3 \cdot \log 5} = \boxed{\textbf{(C) } 6}.</cmath> |
== Solution 2 == | == Solution 2 == | ||
| Line 15: | Line 15: | ||
\end{align*}</cmath> | \end{align*}</cmath> | ||
| − | == Video Solution == | + | == Video Solution by OmegaLearn == |
https://youtu.be/RdIIEhsbZKw?t=605 | https://youtu.be/RdIIEhsbZKw?t=605 | ||
~ pi_is_3.14 | ~ pi_is_3.14 | ||
| + | |||
| + | ==Video Solution (HOW TO THINK CRITICALLY!!!)== | ||
| + | https://youtu.be/FmBU-BT89H4 | ||
| + | |||
| + | ~Education, the Study of Everything | ||
==See Also== | ==See Also== | ||
Latest revision as of 01:35, 28 May 2023
Contents
Problem
What is the value of ![\[\log_37\cdot\log_59\cdot\log_711\cdot\log_913\cdots\log_{21}25\cdot\log_{23}27?\]](http://latex.artofproblemsolving.com/1/0/8/108141fcccb63d051a626897ded91e10882ab0db.png)
Solution 1
From the Change of Base Formula, we have ![\[\frac{\prod_{i=3}^{13} \log (2i+1)}{\prod_{i=1}^{11}\log (2i+1)} = \frac{\log 25 \cdot \log 27}{\log 3 \cdot \log 5} = \frac{(2\log 5)\cdot(3\log 3)}{\log 3 \cdot \log 5} = \boxed{\textbf{(C) } 6}.\]](http://latex.artofproblemsolving.com/3/d/0/3d0aba10cc169dfcc3b353ffe370b5a527574723.png)
Solution 2
Using the chain rule of logarithms 
we get
Video Solution by OmegaLearn
https://youtu.be/RdIIEhsbZKw?t=605
~ pi_is_3.14
Video Solution (HOW TO THINK CRITICALLY!!!)
~Education, the Study of Everything
See Also
| 2018 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 6 |
Followed by Problem 8 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
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