Question

identify the error in the student solution shown below. find the correct answer.
2ln(x) = ln(3x) - ln(9) - 2ln(3)
ln(x²) = ln(3x) - ln(9) - ln(9)
ln(x²) = ln(3x) - 0
ln(x²) = ln(3x/0); division by 0, undefined.

Explanation:

Step1: Analyze Logarithm Subtraction Rule

The student incorrectly applied the logarithm subtraction rule. The rule is $\ln(a) - \ln(b)=\ln(\frac{a}{b})$, but when there's a subtraction of a zero (i.e., $\ln(3x)-0$), it should remain $\ln(3x)$, not $\ln(\frac{3x}{0})$. Division by zero is not involved here as subtracting zero from a logarithm doesn't imply division by zero.

Step2: Simplify the Equation

Start with the correct step after $\ln(x^2)=\ln(3x)-0$. Since subtracting zero doesn't change the value, we have $\ln(x^2)=\ln(3x)$.

Step3: Use Logarithm Equality Property

If $\ln(a)=\ln(b)$, then $a = b$ (for $a,b>0$). So $x^2 = 3x$.

Step4: Solve the Quadratic Equation

Rearrange to $x^2 - 3x = 0$, factor out $x$: $x(x - 3)=0$.

Step5: Find Solutions

Set each factor to zero: $x = 0$ or $x - 3 = 0$. But $x = 0$ is not in the domain of $\ln(x)$, so the solution is $x = 3$.

Answer:

The error was misapplying the logarithm subtraction rule (thinking $\ln(3x)-0$ implies $\ln(\frac{3x}{0})$). The correct answer is $x = 3$.