what is the solution to $\log_{2}(2x^{3}-8)-2\log_{2}x=\log_{2}x$?

$x=1$
$x=2$
$x=3$
$x=4$

Question

what is the solution to $\log_{2}(2x^{3}-8)-2\log_{2}x=\log_{2}x$?

$x=1$
$x=2$
$x=3$
$x=4$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Logarithmic Equations", "Extraneous Solutions of Logarithmic Equations", "Logarithmic Properties" ], "new_concepts": [], "current_concepts": [ "Logarithmic Equations", "Extraneous Solutions of Logarithmic Equations", "Logarithmic Properties" ] } ## Simplify the logarithmic equation $$ \begin{aligned} \log_2(2x^3 - 8) - 2\log_2 x &= \log_2 x \ \log_2(2x^3 - 8) &= 3\log_2 x \ \log_2(2x^3 - 8) &= \log_2(x^3) \end{aligned} $$ ## Solve the resulting algebraic equation $$ \begin{aligned} 2x^3 - 8 &= x^3 \ x^3 &= 8 \ x &= 2 \end{aligned} $$ ## Verify the solution in the original domain $$ \begin{aligned} x > 0 &\implies 2 > 0 \quad (\text{True}) \ 2x^3 - 8 > 0 &\implies 2(2)^3 - 8 = 8 > 0 \quad (\text{True}) \end{aligned} $$

(A) $x = 1$

(B) $x = 2$

(C) $x = 3$

(D) $x = 4$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Logarithmic Equations" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Simplify the logarithmic equation

Solve the resulting algebraic equation

Verify the solution in the original domain

Answer: