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Answer 1 · 2017-09-08T22:32:55

We have #y = 2log (x + y) + 2log (x - y) - log (x^2 + y^2)#

#implies 2log [(x+y)(x-y)] - log (x^2 + y^2)#

#implies 2log (x^2 - y^2) - log (x^2 + y^2)#

#implies log [(x^2 - y^2)^2] - log (x^2 + y^2)#

#implies log [(x^2 - y^2)^2/(x^2 + y^2)]#

Answer 2 · 2017-09-08T22:39:39

#log(((x+y)^2*(x-y)^2)/(x^2+y^2))#

Before starting the answer, the three basic rules for log combinations are as followed:

Now,
#2log(x+y)+2log(x-y)-log(x^2+y^2)#

First we apply the #3^(rd)# rule to the first and second term:

#log(x+y)^2+log(x-y)^2-log(x^2+y^2)#

Then, we apply the first rule to the first two terms

#log((x+y)^2*(x-y)^2)-log(x^2+y^2)#

Finally the third rule is applied

#log(((x+y)^2*(x-y)^2)/(x^2+y^2))#

I hope that this helps. :)