Question

2 ln(x + 2) − ln(−x) = 0

Answer 1

Answer:

$x=-1$

Step-by-step explanation:

Rewrite the equation using the next propierties:

$log(\frac{1}{x} )=-log(x)$

$ylog(x)=log(x^{y} )$

$log(x*y)=log(x)+log(y)$

$ln((x+2)^2)+ln(\frac{-1}{x} )=0\\ln(\frac{(x+2)^2}{-x})=0$

Cancel logarithms by taking exp of both sides:

$\frac{(x+2)^2}{-x} =e^{0} =1$

Multiplying both sides by -x and factoring:

$x^{2} +5x+4=0$

Factoring:

$(x+1)(x+4)=0$

The solutions are:

$x=-1\hspace{3}or\hspace{3}x=-3$

Evaluating x=-4

$2ln(-2)-(4)=0$

This is an absurd because ln(x) is undefined for $x\leq 0$

Evaluating x=-1

$2ln(1)-ln(1)=0\\0-0=0$

Which is correct, hence the solution is:

$x=-1$