2008 AMC 12B Problem 14

Problem 14 of 25IntermediateAlgebra Geometry

A circle has a radius of

\log_{10}(a^2)

and a circumference of

\log_{10}(b^4).

What is

\log_a b?

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Solution

The circumference is

2\pi

times the radius, so

\log_{10}(b^4) = 2\pi \log_{10}(a^2).

Rewriting,

4\log_{10} b = 4\pi \log_{10} a,

hence

\log_{10} b = \pi \log_{10} a.

Therefore

\log_a b = \dfrac{\log_{10} b}{\log_{10} a} = \pi.

Thus, the correct answer is C.

Concepts: logarithm · circumference

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.