2008 AMC 12B Problem 14 — Solution | LIVE by Po-Shen Loh

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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.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

14.

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

$\dfrac{1}{4\pi}$

$\dfrac{1}{\pi}$

$\pi$

$2\pi$

$10^{2\pi}$

Answer: C

Concepts:logarithm circumference

Difficulty rating: 1630

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.

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