Let, log_9(x) = log_12(y) = log_16(x+y) = m(say)

Then, x = 9^m; y = 12 ^m; x+y = 16^m
So, x(x+y) = 144^m = 12^(2m) = y^2
Or, y^2 - xy - x^2 = 0
Or, y/x = (1 +/- sqrt(5))/2 

However, if y/x = (1- sqrt(5))/2 = -0.618. 
If x is positive, then y must be negative, so that log_12(y) is undefined. This is a contradiction.
If y is positive, then x must be negative so that log_9(x) is undefined. This is a contradiction.

Consequently, y/x = (1+ sqrt(5))/2 is the only possible solution to the given problem. 

Edited on February 28, 2008, 5:40 am

Posted by K Sengupta on 2007-03-19 12:01:53