Respuesta :
Answer:
[tex]x = 0.4142[/tex]
Step-by-step explanation:
The first step to solve this equation is placing everything with the logarithmic to one side of the equality, and everything without the exponential to the other side. So
[tex]\ln{x} + \ln{x + 2} = 0[/tex]
Now we have to write the left side as one ln only.
We have that
[tex]\ln{a} + \ln{b} = \ln{a*b}[/tex]
So
[tex]\ln{x} + \ln{x + 2} = 0[/tex]
[tex]\ln{x*(x+2)} = 0[/tex]
[tex]\ln{x^{2} + 2x} = 0[/tex]
We have that the exponential and the ln are inverse functions. This means that [tex]e^{\ln{a}} = a[/tex]. So we apply the exponential to both sides of the equality
[tex]\e^{ln{x^{2} + 2x}} = e^{0}[/tex]
[tex]x^{2} + 2x = 1[/tex]
[tex]x^{2} + 2x - 1 = 0[/tex]
This is a quadratic equation, with roots -2.4142 and 0.4142. There is no ln for negative numbers, so the solution to this equation is:
[tex]x = 0.4142[/tex]
