Alright, let's carefully analyze the steps and identify the error in the student's solution.
### Student's Original Equation:
[tex]\[2 \ln(x) = \ln(3x) - [\ln(9) - 2 \ln(3)]\][/tex]
First, let’s rewrite this by examining each part step-by-step.
### Step-by-Step Solution and Error Identification:
1. Starting Equation:
[tex]\[ 2 \ln(x) = \ln(3x) - [\ln(9) - 2 \ln(3)] \][/tex]
2. The student starts correctly by noting that:
[tex]\[ \ln \left(x^2 \right) = \ln (3x) - [\ln(9) - 2 \ln(3)] \][/tex]
3. Next, simplify [tex]\(\ln(9) - 2 \ln(3)\)[/tex]:
[tex]\[\ln(9) = \ln(3^2) = 2 \ln(3)\][/tex]
So:
[tex]\[\ln(9) - 2 \ln(3) = 2 \ln(3) - 2 \ln(3) = 0\][/tex]
4. Substituting this back into the equation results in:
[tex]\[\ln \left(x^2 \right) = \ln (3x) - 0\][/tex]
[tex]\[\ln \left(x^2 \right) = \ln (3x)\][/tex]
5. Up to this point, everything seems logically correct. However, the student introduces an erroneous step:
[tex]\[\ln (x^2) = \ln \left(\frac{3x}{0}\right)\][/tex]
This step is incorrect because there is no operation that introduces division by zero. The correct step should have been:
[tex]\[\ln \left(x^2\right) = \ln (3x)\][/tex]
6. To find the correct solution, let's use the correct approach:
[tex]\[\ln \left(x^2\right) = \ln (3x)\][/tex]
Since [tex]\(\ln(a) = \ln(b)\)[/tex] implies [tex]\(a = b\)[/tex], we get:
[tex]\[x^2 = 3x\][/tex]
7. Solving [tex]\(x^2 = 3x\)[/tex]:
[tex]\[x^2 - 3x = 0\][/tex]
[tex]\[x(x - 3) = 0\][/tex]
[tex]\[x = 0 \quad \text{or} \quad x = 3\][/tex]
Since [tex]\(x = 0\)[/tex] is not a valid solution in the context of logarithms (logarithm of zero is undefined), we discard this solution, leaving:
[tex]\[x = 3\][/tex]
### Correct Answer:
[tex]\[x = 3\][/tex]
### Detailed Solution Summary:
Inspecting the initial equation and ensuring each simplification step is accurate reveals that the student's final steps introduced division by zero, a clear error. Correct simplification and logical analysis yield the valid solution [tex]\(x = 3\)[/tex].
