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Algebra Difficulty 4.9 AIME Find the answer

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

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

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