HMMT February

Algebra Difficulty 4.9 AIME Find the answer

Which is greater, $\log _{2008}(2009)$ or $\log _{2009}(2010)$ ?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Let $f(x)=\log _{x}(x+1)$ . Then $f^{\prime}(x)=\frac{x \ln x-(x+1) \ln (x+1)}{x(x+1) \ln ^{2} x}<0$ for any $x>1$ , so $f$ is decreasing. Thus $\log _{2008}(2009)$ is greater.

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