Answer:
[tex]x=-1[/tex]
Step-by-step explanation:
Rewrite the equation using the next propierties:
[tex]log(\frac{1}{x} )=-log(x)[/tex]
[tex]ylog(x)=log(x^{y} )[/tex]
[tex]log(x*y)=log(x)+log(y)[/tex]
[tex]ln((x+2)^2)+ln(\frac{-1}{x} )=0\\ln(\frac{(x+2)^2}{-x})=0[/tex]
Cancel logarithms by taking exp of both sides:
[tex]\frac{(x+2)^2}{-x} =e^{0} =1[/tex]
Multiplying both sides by -x and factoring:
[tex]x^{2} +5x+4=0[/tex]
Factoring:
[tex](x+1)(x+4)=0[/tex]
The solutions are:
[tex]x=-1\hspace{3}or\hspace{3}x=-3[/tex]
Evaluating x=-4
[tex]2ln(-2)-(4)=0[/tex]
This is an absurd because ln(x) is undefined for [tex]x\leq 0[/tex]
Evaluating x=-1
[tex]2ln(1)-ln(1)=0\\0-0=0[/tex]
Which is correct, hence the solution is:
[tex]x=-1[/tex]
