To solve the equation [tex]\(\ln (2x + 4) = \ln (x + 3)\)[/tex], we will follow these steps:
1. Understand the property of logarithms:
If [tex]\(\ln A = \ln B\)[/tex], then [tex]\(A = B\)[/tex].
2. Apply the property:
[tex]\[
\ln (2x + 4) = \ln (x + 3) \implies 2x + 4 = x + 3
\][/tex]
3. Solve the linear equation:
[tex]\[
2x + 4 = x + 3
\][/tex]
Subtract [tex]\(x\)[/tex] from both sides:
[tex]\[
2x - x + 4 = 3
\][/tex]
Simplify:
[tex]\[
x + 4 = 3
\][/tex]
Subtract 4 from both sides:
[tex]\[
x = 3 - 4
\][/tex]
Result:
[tex]\[
x = -1
\][/tex]
4. Check the solution:
Substituting [tex]\(x = -1\)[/tex] back into the original equation:
[tex]\[
\ln (2(-1) + 4) = \ln (-1 + 3)
\][/tex]
Simplify the arguments of the logarithms:
[tex]\[
\ln (2 - 2) = \ln 2
\][/tex]
[tex]\[
\ln 2 = \ln 2
\][/tex]
Since both sides are equal, the solution [tex]\(x = -1\)[/tex] is correct.
Therefore, the solution to the equation [tex]\(\ln (2x + 4) = \ln (x + 3)\)[/tex] is [tex]\(x = -1\)[/tex].
The correct answer is:
[tex]\[
\boxed{-1}
\][/tex]
Thus, the answer is option B.
