2009 AMC 12B 第 15 题

Solving each gives $x = \dfrac{\log(\frac{7}{3})}{\log(1 + f(r))}$; a smaller base makes $x$ larger

Compare the values $r,\ \frac{r}{10},\ 2r,\ \sqrt{r},\ \frac{1}{r}$ for $0 \lt r \lt 3$ and find the smallest

Each equation gives $x = \dfrac{\log(\frac{7}{3})}{\log(1 + f(r))},$ which is largest when the positive quantity $f(r)$ is smallest.

For $0 \lt r \lt 3,$ we have $\dfrac r{10}\lt r\lt2r$ and $\dfrac r{10}\lt\sqrt r$ because $r\lt100.$ Also $r^2\lt9\lt10,$ so $\dfrac r{10}\lt\dfrac1r.$ Thus $\frac{r}{10}$ is the smallest of the five quantities, and equation (B) has the largest solution.

Thus, the correct answer is B.