Base e Answers

The error occurs in the last step where the student applied the quotient property of logarithms incorrectly. The term '0' is a scalar value, not a logarithm, so the property ln(a) - ln(b) = ln(a/b) does not apply. Instead, the equation ln(x^2) = ln(3x) - 0 should simply be simplified to ln(x^2) = ln(3x). To find the correct solution: 1. Solve ln(x^2) = ln(3x) by setting the arguments equal: x^2 = 3x. 2. Rearrange the equation: x^2 - 3x = 0. 3. Factor: x(x - 3) = 0. 4. Solutions are x = 0 and x = 3. 5. Check for extraneous solutions: The original equation contains ln(x), and the argument must be positive (x > 0). Therefore, x = 0 is extraneous. 6. The final correct solution is x = 3.