2008 AMC 12B Problem 14

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.