## zmudz one year ago 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 $$a+b.$$

1. freckles

have you tried applying change of base formula

2. freckles

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

3. freckles

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