SOLUTION: Find x and y given that e^x + 3e^y=3 and e^2x - 9e^2y=6, expressing your answers as a logarithm to base e.

Introduce two new variables u =  and v = .

Then your system will be reduced to the system

u   +  3v  = 3,     (1)
u^2 - 9v^2 = 6.     (2)

Factor the second equation

(u-3v)*(u+3v) = 6

and substitute 3 instead of u+3x in the last equation. You will get the system

u   +  3v  = 3,         (1')
u   -  3v  =  = 2.    (2')

Add the two last equations. You will get

2u = 5. Hence, u = .

Now, distract (2) from (1'). You will get 

6v = 1. Hence, v = .

Thus we get 

u =  =  -----> x =  = .

v =  =  -----> y =  = .