Question

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 \(a+b.\)

Answer 1

have you tried applying change of base formula

Answer 2

\[P=\frac{\ln(3)}{\ln(2)} \cdot \frac{\ln(4)}{\ln(3)} \cdot \frac{\ln(5)}{\ln(4)} \cdots \frac{\ln(2008)}{\ln(2007)} \cdot \frac{\ln(2009)}{\ln(2008)}\]
should see a lot of cancellation

Answer 3

then after all that cancellation you can use change of base formula again