document.write( "Question 1109882: solve : (ln(1+4/x)) ^2 +(ln(1-4/(x+4)))^2 =2(ln((3-x)/(x-1)))^2 \n" ); document.write( "

Algebra.Com's Answer #724911 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
The equation has a few restrictions, but we can worry about it later,
\n" ); document.write( "as we may have to get rid of some extraneous solutions, anyway.
\n" ); document.write( "For now, we will assume $\"x%3C%3E0\"$ , $\"x%2B4%3C%3E0\"$ , etc.
\n" ); document.write( " $\"%28ln%281%2B4%2Fx%29%29+%5E2+%2B%28ln%281-4%2F%28x%2B4%29%29%29%5E2\"$ $\"%22=%22\"$ $\"2%28ln%28%283-x%29%2F%28x-1%29%29%29%5E2\"$
\n" ); document.write( " $\"%28ln%28%28x%2B4%29%2Fx%29%29+%5E2+%2B%28ln%28%28x%2B4-4%29%2F%28x%2B4%29%29%29%5E2+\"$ $\"%22=%22\"$ $\"2%28ln%28%283-x%29%2F%28x-1%29%29%29%5E2\"$
\n" ); document.write( " $\"%28ln%28%28x%2B4%29%2Fx%29%29+%5E2+%2B%28ln%28x%2F%28x%2B4%29%29%29%5E2+\"$ $\"%22=%22\"$ $\"2%28ln%28%283-x%29%2F%28x-1%29%29%29%5E2\"$
\n" ); document.write( "

$\"%22=%22\"$ $\"2%28ln%28%283-x%29%2F%28x-1%29%29%29%5E2\"$
\n" ); document.write( " $\"%28ln%28%28x%2B4%29%2Fx%29+%2Bln%28x%2F%28x%2B4%29%29%29%5E2\"$ $\"%22=%22\"$

\n" ); document.write( " $\"%28ln%28%28x%2B4%29%2Fx%29%2A%28x%2F%28x%2B4%29%29%29%5E2\"$ $\"%22=%22\"$

\n" ); document.write( " $\"%28ln%281%29%29%5E2\"$ $\"%22=%22\"$

\n" ); document.write( " $\"0\"$ $\"%22=%22\"$

\n" ); document.write( " $\"%28ln%28%283-x%29%2F%28x-1%29%29%29%5E2%2Bln%28%28x%2B4%29%2Fx%29ln%28x%2F%28x%2B4%29%29\"$ $\"%22=%22\"$ $\"0\"$
\n" ); document.write( " $\"%28ln%28%283-x%29%2F%28x-1%29%29%29%5E2%29\"$ $\"%22=%22\"$ $\"-ln%28%28x%2B4%29%2Fx%29ln%28x%2F%28x%2B4%29%29\"$
\n" ); document.write( " $\"%28ln%28%283-x%29%2F%28x-1%29%29%29%5E2%29\"$ $\"%22=%22\"$ $\"ln%28x%2F%28x%2B4%29%29ln%28x%2F%28x%2B4%29%29\"$
\n" ); document.write( " $\"%28ln%28%283-x%29%2F%28x-1%29%29%29%5E2%29\"$ $\"%22=%22\"$ $\"%28ln%28x%2F%28x%2B4%29%29%29%5E2\"$
\n" ); document.write( "If both logarithms have the same sign (both positive or both negative,
\n" ); document.write( " $\"ln%28%283-x%29%2F%28x-1%29%29%29\"$ $\"%22=%22\"$ $\"ln%28x%2F%28x%2B4%29%29\"$ <--> $\"%283-x%29%2F%28x-1%29\"$ $\"%22=%22\"$ $\"x%2F%28x%2B4%29\"$ .
\n" ); document.write( "Otherwise,
\n" ); document.write( " $\"ln%28%283-x%29%2F%28x-1%29%29%29\"$ $\"%22=%22\"$ $\"-ln%28x%2F%28x%2B4%29%29\"$ <--> $\"ln%28%283-x%29%2F%28x-1%29%29%29\"$ $\"%22=%22\"$ $\"-ln%28%28x%2B4%29%2Fx%29\"$ <--> $\"%283-x%29%2F%28x-1%29\"$ $\"%22=%22\"$ $\"%28x%2B4%29%2Fx\"$ .
\n" ); document.write( "
\n" ); document.write( "And we trudge on.
\n" ); document.write( "
\n" ); document.write( " $\"%283-x%29%2F%28x-1%29\"$ $\"%22=%22\"$ $\"x%2F%28x%2B4%29\"$
\n" ); document.write( " $\"%283-x%29%28x%2B4%29\"$ $\"%22=%22\"$ $\"x%28x-1%29\"$
\n" ); document.write( " $\"12-x-x%5E2=x%5E2-x\"$
\n" ); document.write( " $\"12-x%5E2=x%5E2\"$
\n" ); document.write( " $\"12=2x%5E2\"$
\n" ); document.write( " $\"x%5E2=6\"$
\n" ); document.write( " $\"x=%22+%22+%2B-+sqrt%286%29\"$
\n" ); document.write( "For $\"highlight%28x=sqrt%286%29%29\"$ ,
\n" ); document.write( " $\"3-x\"$ , $\"x-1\"$ , $\"x\"$ , $\"x%2B4\"$ , $\"%283-x%29%2F%28x-1%29\"$ , and $\"x%2F%28x%2B4%29\"$ are all positive,
\n" ); document.write( "so it looks like a valid solution, and it verifies.
\n" ); document.write( "For $\"x=-sqrt%286%29\"$ , $\"x%2F%28x%2B4%29%3C0\"$ ,
\n" ); document.write( "so $\"ln%281-4%2F%28x%2B4%29%29=ln%28x%2F%28x%2B4%29%29\"$ does not exist/is undefined,
\n" ); document.write( "so we know $\"x=-sqrt%286%29\"$ is not a solution.
\n" ); document.write( "I do not have to try to verify that.
\n" ); document.write( "
\n" ); document.write( " $\"%283-x%29%2F%28x-1%29\"$ $\"%22=%22\"$ $\"%28x%2B4%29%2Fx\"$
\n" ); document.write( " $\"%283-x%29x\"$ $\"%22=%22\"$ $\"%28x%2B4%29%28x-1%29\"$
\n" ); document.write( " $\"3x-x%5E2=x%5E2%2B3x-4\"$
\n" ); document.write( " $\"-x%5E2=x%5E2-4\"$
\n" ); document.write( " $\"2x%5E2=4\"$
\n" ); document.write( " $\"x%5E2=2\"$
\n" ); document.write( " $\"x=%22+%22+%2B-+sqrt%282%29\"$
\n" ); document.write( "For $\"highlight%28x=sqrt%282%29%29\"$ ,
\n" ); document.write( " $\"3-x\"$ , $\"x-1\"$ , $\"x\"$ , $\"x%2B4\"$ , $\"%283-x%29%2F%28x-1%29\"$ , and $\"x%2F%28x%2B4%29\"$ are all positive,
\n" ); document.write( "so it looks like a valid solution, and it verifies.
\n" ); document.write( "For $\"x=-sqrt%282%29\"$ , $\"x%2F%28x%2B4%29%3C0\"$ ,
\n" ); document.write( "so $\"ln%281-4%2F%28x%2B4%29%29=ln%28x%2F%28x%2B4%29%29\"$ does not exist/is undefined,
\n" ); document.write( "so we know $\"x=-sqrt%282%29\"$ is not a solution.
\n" ); document.write( "I do not have to try to verify that.
\n" ); document.write( "
\n" ); document.write( "HOW WE VERIFY:
\n" ); document.write( "For $\"highlight%28x=sqrt%282%29%29\"$
\n" ); document.write( " $\"%283-x%29%2F%28x-1%29\"$ $\"%22=%22\"$ $\"%283-sqrt%282%29%29%2F%28sqrt%282%29-1%29\"$ $\"%22=%22\"$ $\"%283-sqrt%282%29%29%28sqrt%282%29%2B1%29%2F%28%28sqrt%282%29-1%29%28sqrt%282%29%2B1%29%29\"$ $\"%22=%22\"$ $\"%283sqrt%282%29%2B3-2-sqrt%282%29%29%2F%282-1%29%29\"$ $\"%22=%22\"$ $\"2sqrt%282%29%2B1\"$
\n" ); document.write( "
\n" ); document.write( " $\"%28x%2B4%29%2Fx\"$ $\"%22=%22\"$ $\"%28sqrt%282%29%2B4%29%2Fsqrt%282%29\"$ $\"%22=%22\"$ $\"%28sqrt%282%29%2B4%29sqrt%282%29%2F%28sqrt%282%29sqrt%282%29%29\"$ $\"%22=%22\"$ $\"%282%2B4sqrt%282%29%29%2F2\"$ $\"%22=%22\"$ $\"2sqrt%282%29%2B1\"$ \n" ); document.write( "

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