To solve the equation [tex]\(\ln(2x + 4) = \ln(x + 3)\)[/tex], we will use properties of logarithms.
### Step-by-Step Solution
1. Starting Equation:
[tex]\[
\ln(2x + 4) = \ln(x + 3)
\][/tex]
2. Property of Logarithms:
Since the natural logarithm function is one-to-one, if [tex]\(\ln A = \ln B\)[/tex], then [tex]\(A = B\)[/tex]. Therefore, we can equate the arguments of the logarithms:
[tex]\[
2x + 4 = x + 3
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
To isolate [tex]\(x\)[/tex], we will subtract [tex]\(x\)[/tex] from both sides of the equation:
[tex]\[
2x + 4 - x = x + 3 - x
\][/tex]
Simplifying both sides, we get:
[tex]\[
x + 4 = 3
\][/tex]
4. Further Simplifying:
Subtract 4 from both sides:
[tex]\[
x + 4 - 4 = 3 - 4
\][/tex]
[tex]\[
x = -1
\][/tex]
5. Verify the Solution:
Substitute [tex]\(x = -1\)[/tex] back into the original equation to ensure it is correct:
[tex]\[
\ln(2(-1) + 4) = \ln((-1) + 3)
\][/tex]
[tex]\[
\ln(2 - 2) = \ln(2)
\][/tex]
[tex]\[
\ln(2) = \ln(2)
\][/tex]
Both sides are equal, confirming that [tex]\(x = -1\)[/tex] is indeed a solution.
### Conclusion
The solution to the equation [tex]\(\ln(2x + 4) = \ln(x + 3)\)[/tex] is:
[tex]\[
\boxed{x = -1}
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{\text{B. } x = -1}
\][/tex]
