To find the potential solutions for the equation
[tex]$
2 \ln (x+3) = 0,
$[/tex]
let's break down the equation step by step.
1. Simplify the equation:
[tex]\[
2 \ln (x+3) = 0
\][/tex]
Divide both sides of the equation by 2:
[tex]\[
\ln (x+3) = 0
\][/tex]
2. Solve for [tex]\(x+3\)[/tex]:
Recall the definition of the natural logarithm: [tex]\(\ln (a)=0\)[/tex] if and only if [tex]\(a=1\)[/tex]. Therefore, we can set up the following equation:
[tex]\[
x+3 = 1
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Subtract 3 from both sides:
[tex]\[
x = 1 - 3
\][/tex]
[tex]\[
x = -2
\][/tex]
The solution to the equation [tex]\(2 \ln (x+3) = 0\)[/tex] is [tex]\(x = -2\)[/tex].
Now, let's verify if [tex]\(x = -2\)[/tex] is included in the provided options and check if there's any other solution:
Given choices are:
- [tex]\( x = -3 \)[/tex] and [tex]\( x = -4 \)[/tex]
- [tex]\( x = -2 \)[/tex] and [tex]\( x = -4 \)[/tex]
- [tex]\( x = 2 \)[/tex] and [tex]\( x = -3 \)[/tex]
- [tex]\( x = 2 \)[/tex] and [tex]\( x = 4 \)[/tex]
Out of these options, the choice [tex]\( x = -2 \)[/tex] and [tex]\( x = -4 \)[/tex] is the one that includes [tex]\( x = -2 \)[/tex]. However, there can't be two solutions to this specific logarithmic equation since it's a straightforward equation with a single logarithmic term.
So, based on the provided answers, the correct solution is:
[tex]\[ \boxed{x = -2} \][/tex]
But since we must choose from the provided options and [tex]\( x = -2 \)[/tex] is present only in the option [tex]\( x = -2 \)[/tex] and [tex]\( x = -4 \)[/tex], we conclude that this is the closest matching option, although [tex]\( x = -4 \)[/tex] is extraneous.
Thus, the answer based on the given options is:
[tex]\[ x = -2 \text{ and } x = -4 \][/tex]
