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HMMT February

Algebra Difficulty 4.7 AIME Find the answer

Find log⁡n(12)log⁡n−1(13)⋯log⁡2(1n)\log _{n}\left(\frac{1}{2}\right) \log _{n-1}\left(\frac{1}{3}\right) \cdots \log _{2}\left(\frac{1}{n}\right) in terms of nn.

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

Solution

Using log⁡1x=−log⁡x\log \frac{1}{x}=-\log x and log⁡ba=log⁡alog⁡b\log _{b} a=\frac{\log a}{\log b}, we get that the product equals (−log⁡2)(−log⁡3)⋯(−log⁡n)log⁡n⋯log⁡3log⁡2=(−1)n−1\frac{(-\log 2)(-\log 3) \cdots(-\log n)}{\log n \cdots \log 3 \log 2}=(-1)^{n-1}.

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