Question

identify the error in the student solution shown below. find the correct answer.

\\(2\ln(x) = \ln(3x) - \ln(9) - 2\ln(3)\\)
\\(\ln(x^2) = \ln(3x) - \ln(9) - \ln(9)\\)
\\(\ln(x^2) = \ln(3x) - 0\\)
\\(\ln(x^2) = \ln(3x/0)\\): division by 0, undefined.

Explanation:

Identify the student's error

Using the Logarithmic Quotient Rule and Logarithmic Power Rule knowledge points
The student wrote:

$$ \ln(x^2) = \ln(3x) - 0 \implies \ln(x^2) = \ln(3x/0) $$

This is incorrect because subtracting \(0\) from a term does not change its value:

$$ \ln(3x) - 0 = \ln(3x) $$

The student incorrectly applied the quotient rule \(\ln(A) - \ln(B) = \ln(A/B)\) by treating \(0\) as \(\ln(0)\), which is undefined, or by dividing the argument by \(0\).

Solve the equation correctly

Using the Solving Logarithmic Equations knowledge point

$$
LATEXBLOCK0
$$

Solve for x

Using the Solving Logarithmic Equations knowledge point

$$
LATEXBLOCK1
$$

Verify the solutions

Using the Solving Logarithmic Equations knowledge point
The original equation contains \(\ln(x)\) and \(\ln(3x)\). The domain of the natural logarithm requires the argument to be strictly positive:

$$ x > 0 $$

Therefore, \(x = 0\) is an extraneous solution, and the only valid solution is:

$$ x = 3 $$

Answer:

Error:
In the last step, the student incorrectly simplified \(\ln(3x) - 0\) as \(\ln(3x/0)\). Subtracting \(0\) from any expression leaves it unchanged, so \(\ln(3x) - 0 = \ln(3x)\).

Correct Solution:

$$x = 3$$