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.
+128475 Note ...using the change-of-base theorem we can write
P = log2 3 * log 3 4 * log 4 5 * ....... *log 2007 2008 * log 2008 2009 as
log 3 * log 4 * log 5 log 2008 * log 2009
P = _____ _____ _____ * ...... * _______ _________ =
log 2 log 3 log 4 . log 2007 log 2008
1 log 2009
____ * ________ =
log 2 1
log 2009
________ = log 2 2009 = log a b
log 2
a = 2 b = 2009
a + b =
2011
