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Select the correct answer.

What is the solution to this equation?

[tex]\[ \ln (2x + 4) = \ln (x + 3) \][/tex]

A. [tex]\( x = 7 \)[/tex]

B. [tex]\( x = -1 \)[/tex]

C. [tex]\( x = -7 \)[/tex]

D. [tex]\( x = 1 \)[/tex]


Sagot :

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]
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