Difference between revisions of "2008 AMC 12B Problems/Problem 14"

Latest revision as of 09:54, 4 July 2013

A circle has a radius of $\log_{10}{(a^2)}$ and a circumference of $\log_{10}{(b^4)}$ . What is $\log_{a}{b}$ ?

$\textbf{(A)}\ \frac{1}{4\pi} \qquad \textbf{(B)}\ \frac{1}{\pi} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ 2\pi \qquad \textbf{(E)}\ 10^{2\pi}$

Let $C$ be the circumference of the circle, and let $r$ be the radius of the circle.

Using log properties, $C=\log_{10}{(b^4)}=4\log_{10}{(b)}$ and $r=\log_{10}{(a^2)}=2\log_{10}{(a)}$ .

Since $C=2\pi r$ , $4\log_{10}{(b)}=2\pi\cdot2\log_{10}{(a)} \Rightarrow \log_{a}{b} = \frac{\log_{10}{(b)}}{\log_{10}{(a)}}=\pi \Rightarrow C$ .

2008 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
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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 ==See Also==
 {{AMC12 box|year=2008|ab=B|num-b=13|num-a=15}}
+{{MAA Notice}}