2018 AMC 12B Problems/Problem 7

Problem

What is the value of $\log_37\cdot\log_59\cdot\log_711\cdot\log_913\cdots\log_{21}25\cdot\log_{23}27?$ $\textbf{(A) } 3 \qquad \textbf{(B) } 3\log_{7}23 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 9 \qquad \textbf{(E) } 10$

Solution 1

From the Change of Base Formula, we have $\frac{\prod_{i=3}^{13} \log (2i+1)}{\prod_{i=1}^{11}\log (2i+1)} = \frac{\log 25 \cdot \log 27}{\log 3 \cdot \log 5} = \frac{(2\log 5)\cdot(3\log 3)}{\log 3 \cdot \log 5} = \boxed{\textbf{(C) } 6}.$

Solution 2

Using the chain rule of logarithms $\log _{a} b \cdot \log _{b} c = \log _{a} c,$ we get $\begin{align*} \log_37\cdot\log_59\cdot\log_711\cdot\log_913\cdots\log_{21}25\cdot\log_{23}27 &= (\log _{3} 7 \cdot \log _{7} 11 \cdots \log _{23} 27) \cdot (\log _{5} 9 \cdot \log _{9} 13 \cdots \log _{21} 25) \\ &= \log _{3} 27 \cdot \log _{5} 25 \\ &= 3 \cdot 2 \\ &= \boxed{\textbf{(C) } 6}. \end{align*}$

Solution 3

Using the Change of Base Formula, we have $\frac{\log(7) \cdot \log(9)\cdot \log(11) ... \log(25) \cdot \log(27)}{\log(3) \cdot \log(5)\cdot \log(7)...\log(21) \cdot \log(23)}$

From this, we realize that the top logs from $\log(7)$ - $\log(23)$ cancel with the bottom logs from $\log(7)$ - $\log(23)$ . This leaves us with $\frac{\log(25) \cdot \log(27)}{\log(3) \cdot \log(5)}$ . Going from this step, we realize that we can use the communicative property to achieve $\frac{\log(25) \cdot \log(27)}{\log(5) \cdot \log(3)}$ .

Therefore, we can use the Change of Base Formula to get back to $\log_525 \cdot \log_327$ . This equals to $2 \cdot 5 = \boxed{\textbf{(C) } 6}$ .

~Dirextrixxx

Video Solution

https://youtu.be/RdIIEhsbZKw?t=605

~ pi_is_3.14

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

Page

Toolbox

Search