a student solved $\log_{4}(2x - 12) = 3$, as shown.
step 1: $2x - 12 = 3^4$
step 2: $2x - 12 = 81$
step 4: $2x = 93$
step 5: $x = 46.5$
explain the error and find the correct solution.

Question

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Solving Logarithmic Equations" ], "new_concepts": [], "current_concepts": [ "Solving Logarithmic Equations", "Logarithmic Evaluation" ] } ## Identify the student's error Using the *Solving Logarithmic Equations* knowledge point $$ \begin{aligned} &\log_{4}(2x - 12) = 3 \implies 2x - 12 = 4^3\ &\text{Student's Step 1: } 2x - 12 = 3^4\ &\text{Error: The student swapped the base and the exponent.} \end{aligned} $$ ## Solve the equation correctly Using the *Solving Logarithmic Equations* knowledge point $$ \begin{aligned} &2x - 12 = 4^3\ &2x - 12 = 64\ &2x = 76\ &x = 38 \end{aligned} $$ The student made an error in Step 1 by swapping the base and the exponent when rewriting the logarithmic equation in exponential form. Instead of raising the base $4$ to the power of $3$ ($4^3$), they raised $3$ to the power of $4$ ($3^4$). The correct solution is: $$ \begin{aligned} \log_{4}(2x - 12) &= 3 \ 2x - 12 &= 4^3 \ 2x - 12 &= 64 \ 2x &= 76 \ x &= 38 \end{aligned} $$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Solving Logarithmic Equations" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Identify the student's error

Solve the equation correctly

Answer: