Select the correct answer.

What is the solution to this equation?
[tex]\[ \ln (2x + 4) = \ln (x + 3) \][/tex]

A. [tex]\( x = 1 \)[/tex]
B. [tex]\( x = -1 \)[/tex]
C. [tex]\( x = -7 \)[/tex]
D. [tex]\( x = 7 \)[/tex]



Answer :

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.