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2008 AMC 12B Problems/Problem 14

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Problem

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

$\textbf{(A)}\ \frac{1}{4\pi} \qquad \textbf{(B)}\ \frac{1}{\pi} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ 2\pi \qquad \te...$

Solution

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

See Also

2008 AMC 12B (Problems • 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

Retrieved from "http://www.artofproblemsolving.com/Wiki/index.php/2008_AMC_12B_Problems/Problem_14"

Looking for a challenging geometry text? Preparing for MATHCOUNTS or the AMC exams? Check out Art of Problem Solving's Introduction to Geometry by Richard Rusczyk.

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