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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
 #1
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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

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