Difference between revisions of "2002 AMC 12B Problems/Problem 22"

Latest revision as of 17:26, 13 January 2022

Problem

For all integers $n$ greater than $1$ , define $a_n = \frac{1}{\log_n 2002}$ . Let $b = a_2 + a_3 + a_4 + a_5$ and $c = a_{10} + a_{11} + a_{12} + a_{13} + a_{14}$ . Then $b- c$ equals

$\mathrm{(A)}\ -2 \qquad\mathrm{(B)}\ -1 \qquad\mathrm{(C)}\ \frac{1}{2002} \qquad\mathrm{(D)}\ \frac{1}{1001} \qquad\mathrm{(E)}\ \frac 12$

Solution

By the change of base formula, $a_n = \frac{1}{\frac{\log 2002}{\log n}} = \left(\frac{1}{\log 2002}\right) \log n$ . Thus $\begin{align*}b- c &= \left(\frac{1}{\log 2002}\right)(\log 2 + \log 3 + \log 4 + \log 5 - \log 10 - \log 11 - \log 12 - \log 13 - \log 14)\\ &= \left(\frac{1}{\log 2002}\right)\left(\log \frac{2 \cdot 3 \cdot 4 \cdot 5}{10 \cdot 11 \cdot 12 \cdot 13 \cdot 14}\right)\\ &= \left(\frac{1}{\log 2002}\right) \log 2002^{-1} = -\left(\frac{\log 2002}{\log 2002}\right) = -1 \Rightarrow \mathrm{(B)}\end{align*}$

Solution 2

Note that $\frac{1}{\log_a b}=\log_b a$ . Thus $a_n=\log_{2002} n$ . Also notice that if we have a log sum, we multiply, and if we have a log product, we divide. Using these properties, we get that the result is the following:

$\log_{2002}\left(\frac{2*3*4*5}{10*11*12*13*14}=\frac{1}{11*13*14}=\frac{1}{2002}\right)=\boxed{\textbf{(B)}-1}$

~yofro

Solution 3

Note that $a_2 = \frac{1}{\log_2 2002}$ . $1$ is also equal to $\log_2 2$ . So $a_2 = \frac{\log_2 2}{\log_2 2002}$ . By the change of bases formula, $a_2 = \log_{2002} 2$ . Following the same reasoning, $a_3 = \log_{2002} 3$ , $a_4 = \log_{2002} 4$ and so on.

$b = \log_{2002} 2 + \log_{2002} 3 + .....+ \log_{2002} 5 = \log_{2002} 5! = \log_{2002} 120$ Now solving for $c$ , we see that it equals $\log_{2002} (10\cdot 11 \cdot 12 \cdot 13 \cdot 14)$ $b-c = \log_{2002} 120 - \log_{2002} 240240 \rightarrow \log_{2002} \frac{1}{2002} = \boxed{-1}$

~YBSuburbanTea

@@ Line 7: / Line 7: @@
 \qquad\mathrm{(D)}\ \frac{1}{1001}
 \qquad\mathrm{(E)}\ \frac 12</math>
 == Solution ==
-=== Solution 1 ===
 By the [[Logarithms#Logarithmic Properties|change of base]] formula, <math>a_n = \frac{1}{\frac{\log 2002}{\log n}} = \left(\frac{1}{\log 2002}\right) \log n </math>. Thus
 <cmath>\begin{align*}b- c &= \left(\frac{1}{\log 2002}\right)(\log 2 + \log 3 + \log 4 + \log 5 - \log 10 - \log 11 - \log 12 - \log 13 - \log 14)\\
@@ Line 14: / Line 15: @@
 &= \left(\frac{1}{\log 2002}\right) \log 2002^{-1} = -\left(\frac{\log 2002}{\log 2002}\right) = -1 \Rightarrow \mathrm{(B)}\end{align*}</cmath>
-=== Solution 2 ===
+== Solution 2 ==
+Note that <math>\frac{1}{\log_a b}=\log_b a</math>. Thus <math>a_n=\log_{2002} n</math>. Also notice that if we have a log sum, we multiply, and if we have a log product, we divide. Using these properties, we get that the result is the following:
+<cmath>\log_{2002}\left(\frac{2*3*4*5}{10*11*12*13*14}=\frac{1}{11*13*14}=\frac{1}{2002}\right)=\boxed{\textbf{(B)}-1}</cmath>
+~yofro
+== Solution 3 ==
+Note that <math>a_2 = \frac{1}{\log_2 2002}</math>. <math>1</math> is also equal to <math>\log_2 2</math>. So <math>a_2 = \frac{\log_2 2}{\log_2 2002}</math>. By the change of bases formula, <math>a_2 = \log_{2002} 2</math>. Following the same reasoning, <math>a_3 = \log_{2002} 3</math>, <math>a_4 = \log_{2002} 4</math> and so on.
+<cmath>b = \log_{2002} 2 + \log_{2002} 3 + .....+ \log_{2002} 5 = \log_{2002} 5! = \log_{2002} 120</cmath>
+Now solving for <math>c</math>, we see that it equals <math>\log_{2002} (10\cdot 11 \cdot 12 \cdot 13 \cdot 14)</math>
+<cmath>b-c = \log_{2002} 120 - \log_{2002} 240240 \rightarrow \log_{2002} \frac{1}{2002} = \boxed{-1}</cmath>
-<math>a_n = \frac{1}{\log_n 2002} = \frac{\log_n n}{log_n 2002} = \log_{2002} {n}</math> So
+~YBSuburbanTea
-<cmath>\begin{align*}
-b- c &= \left(log_2002 2 + log_2002 3 + log_2002 4 + log_2002 5 - log_2002 10 - log_2002 11 - log_2002 12 - log_2002 13 - log_2002 14))\\
-&= \left(log_2002 (\frac{2 \cdot 3 \cdot 4 \cdot 5}{10 \cdot 11 \cdot 12 \cdot 13 \cdot 14}))\\
-&= \left(log_2002 2002^-1) = -1 \Rightarrow \mathrm{(B)}
-\end{align*}</cmath>
 == See also ==

2002 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 21	Followed by Problem 23
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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