2008 AMC 12B Problem 14

Attempt Problem 14 of the 2008 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2008 AMC 12B solutions, or check the answer key.

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14.

A circle has a radius of

\log_{10}(a^2)

and a circumference of

\log_{10}(b^4).

What is

\log_a b?

\dfrac{1}{4\pi}

\dfrac{1}{\pi}

\pi

2\pi

10^{2\pi}

Circumference equals

2\pi

times the radius

Use

\log_{10}(a^2) = 2\log_{10} a

and

\log_{10}(b^4) = 4\log_{10} b,

then form

\dfrac{\log_{10} b}{\log_{10} a}

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.

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