4. solve for $x$ : $4^{x-1} = 27$

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Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Exponential Equations", "Logarithmic Conversion" ], "current_concepts": [ "Exponential Equations", "Logarithmic Conversion" ] } ## Apply logarithms to both sides $$ \log(4^{x-1}) = \log(27) $$ ## Use power property of logarithms $$ (x-1)\log(4) = \log(27) $$ ## Solve for x $$ \begin{aligned} x-1 &= \frac{\log(27)}{\log(4)} \ x &= 1 + \frac{\log(3^3)}{\log(2^2)} \ x &= 1 + \frac{3\log(3)}{2\log(2)} \ x &= 1 + \frac{3}{2}\log_2(3) \end{aligned} $$ $x = 1 + \frac{3\log(3)}{2\log(2)}$ or $x = 1 + \frac{3}{2}\log_2(3)$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Exponential Equations" ] }

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Question

Explanation:

Apply logarithms to both sides

Use power property of logarithms

Solve for x

Answer: