SOLUTION: solve : (ln(1+4/x)) ^2 +(ln(1-4/(x+4)))^2 =2(ln((3-x)/(x-1)))^2

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Question 1109882: solve : (ln(1+4/x)) ^2 +(ln(1-4/(x+4)))^2 =2(ln((3-x)/(x-1)))^2
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
The equation has a few restrictions, but we can worry about it later,
as we may have to get rid of some extraneous solutions, anyway.
For now, we will assume x%3C%3E0 , x%2B4%3C%3E0 , etc.
%28ln%281%2B4%2Fx%29%29+%5E2+%2B%28ln%281-4%2F%28x%2B4%29%29%29%5E2 %22=%22 2%28ln%28%283-x%29%2F%28x-1%29%29%29%5E2
%28ln%28%28x%2B4%29%2Fx%29%29+%5E2+%2B%28ln%28%28x%2B4-4%29%2F%28x%2B4%29%29%29%5E2+ %22=%22 2%28ln%28%283-x%29%2F%28x-1%29%29%29%5E2
%28ln%28%28x%2B4%29%2Fx%29%29+%5E2+%2B%28ln%28x%2F%28x%2B4%29%29%29%5E2+ %22=%22 2%28ln%28%283-x%29%2F%28x-1%29%29%29%5E2
%22=%22 2%28ln%28%283-x%29%2F%28x-1%29%29%29%5E2
%28ln%28%28x%2B4%29%2Fx%29+%2Bln%28x%2F%28x%2B4%29%29%29%5E2 %22=%22
%28ln%28%28x%2B4%29%2Fx%29%2A%28x%2F%28x%2B4%29%29%29%5E2 %22=%22
%28ln%281%29%29%5E2 %22=%22
0 %22=%22
%28ln%28%283-x%29%2F%28x-1%29%29%29%5E2%2Bln%28%28x%2B4%29%2Fx%29ln%28x%2F%28x%2B4%29%29 %22=%22 0
%28ln%28%283-x%29%2F%28x-1%29%29%29%5E2%29 %22=%22 -ln%28%28x%2B4%29%2Fx%29ln%28x%2F%28x%2B4%29%29
%28ln%28%283-x%29%2F%28x-1%29%29%29%5E2%29 %22=%22 ln%28x%2F%28x%2B4%29%29ln%28x%2F%28x%2B4%29%29
%28ln%28%283-x%29%2F%28x-1%29%29%29%5E2%29 %22=%22 %28ln%28x%2F%28x%2B4%29%29%29%5E2
If both logarithms have the same sign (both positive or both negative,
ln%28%283-x%29%2F%28x-1%29%29%29 %22=%22 ln%28x%2F%28x%2B4%29%29 <--> %283-x%29%2F%28x-1%29 %22=%22 x%2F%28x%2B4%29 .
Otherwise,
ln%28%283-x%29%2F%28x-1%29%29%29 %22=%22 -ln%28x%2F%28x%2B4%29%29 <--> ln%28%283-x%29%2F%28x-1%29%29%29 %22=%22 -ln%28%28x%2B4%29%2Fx%29 <--> %283-x%29%2F%28x-1%29 %22=%22 %28x%2B4%29%2Fx .

And we trudge on.

%283-x%29%2F%28x-1%29 %22=%22 x%2F%28x%2B4%29
%283-x%29%28x%2B4%29 %22=%22 x%28x-1%29
12-x-x%5E2=x%5E2-x
12-x%5E2=x%5E2
12=2x%5E2
x%5E2=6
x=%22+%22+%2B-+sqrt%286%29
For highlight%28x=sqrt%286%29%29 ,
3-x , x-1 , x , x%2B4 , %283-x%29%2F%28x-1%29 , and x%2F%28x%2B4%29 are all positive,
so it looks like a valid solution, and it verifies.
For x=-sqrt%286%29 , x%2F%28x%2B4%29%3C0 ,
so ln%281-4%2F%28x%2B4%29%29=ln%28x%2F%28x%2B4%29%29 does not exist/is undefined,
so we know x=-sqrt%286%29 is not a solution.
I do not have to try to verify that.

%283-x%29%2F%28x-1%29 %22=%22 %28x%2B4%29%2Fx
%283-x%29x %22=%22 %28x%2B4%29%28x-1%29
3x-x%5E2=x%5E2%2B3x-4
-x%5E2=x%5E2-4
2x%5E2=4
x%5E2=2
x=%22+%22+%2B-+sqrt%282%29
For highlight%28x=sqrt%282%29%29 ,
3-x , x-1 , x , x%2B4 , %283-x%29%2F%28x-1%29 , and x%2F%28x%2B4%29 are all positive,
so it looks like a valid solution, and it verifies.
For x=-sqrt%282%29 , x%2F%28x%2B4%29%3C0 ,
so ln%281-4%2F%28x%2B4%29%29=ln%28x%2F%28x%2B4%29%29 does not exist/is undefined,
so we know x=-sqrt%282%29 is not a solution.
I do not have to try to verify that.

HOW WE VERIFY:
For highlight%28x=sqrt%282%29%29
%283-x%29%2F%28x-1%29 %22=%22 %283-sqrt%282%29%29%2F%28sqrt%282%29-1%29 %22=%22 %283-sqrt%282%29%29%28sqrt%282%29%2B1%29%2F%28%28sqrt%282%29-1%29%28sqrt%282%29%2B1%29%29 %22=%22 %283sqrt%282%29%2B3-2-sqrt%282%29%29%2F%282-1%29%29 %22=%22 2sqrt%282%29%2B1

%28x%2B4%29%2Fx %22=%22 %28sqrt%282%29%2B4%29%2Fsqrt%282%29 %22=%22 %28sqrt%282%29%2B4%29sqrt%282%29%2F%28sqrt%282%29sqrt%282%29%29 %22=%22 %282%2B4sqrt%282%29%29%2F2 %22=%22 2sqrt%282%29%2B1