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# Let where and ​ and are relatively prime positive integers. Find the smallest possible value of

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Let $$P = \log_a b,$$ where $$P = \log_2 3 \cdot \log_3 4 \cdot \log_4 5 \cdots \log_{2008} 2009$$ and $$a$$ and $$b$$ are relatively prime positive integers. Find the smallest possible value of $$a+b.$$

Feb 23, 2021

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The product is 11.97226185..., or $\log_2 4018$.  So a + b = 2 + 4018 = 4020.

Feb 23, 2021
#2
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P ==10.97226185

10.97226185 = Log_a(b)

b =a^10.97226185

a ==2   and   b==2009

Feb 23, 2021