Berkeley Math Circle

Algebra Difficulty 4.3 AIME Prove it United States

Problem:
Determine, with proof, the value of $\log_{2} 3 \log_{3} 4 \log_{4} 5 \ldots \log_{255} 256$ .

Solution

Solution:
We use the fact that $\log_{b} a = \frac{\log a}{\log b}$ . Thus, the product equals
$\frac{\log 3}{\log 2} \frac{\log 4}{\log 3} \ldots \frac{\log 256}{\log 255} = \frac{\log 256}{\log 2} = \log_{2}(256) = 8$

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