which of the following shows the true solution to the logarithmic equation $3\log_{2}(2x) = 3$?

$x = -1$
$x = 1$
$x = -1$ and $x = 1$
$x = 0$, $x = -1$, and $x = 1$

Question

which of the following shows the true solution to the logarithmic equation $3\log_{2}(2x) = 3$?

$x = -1$
$x = 1$
$x = -1$ and $x = 1$
$x = 0$, $x = -1$, and $x = 1$

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" ], "new_concepts": [], "current_concepts": [ "Logarithmic Equations", "Extraneous Solutions of Logarithmic Equations" ] } ## Isolate the logarithmic term $$ \begin{aligned} 3\log_2(2x) &= 3 \ \log_2(2x) &= 1 \end{aligned} $$ ## Convert to exponential form and solve $$ \begin{aligned} 2x &= 2^1 \ 2x &= 2 \ x &= 1 \end{aligned} $$ ## Verify the solution $$ \begin{aligned} \text{For } x = 1: \quad 2x = 2(1) = 2 > 0 \quad (\text{valid domain}) \ 3\log_2(2(1)) = 3\log_2(2) = 3(1) = 3 \end{aligned} $$

$x = -1$

$x = 1$

$x = -1$ and $x = 1$

$x = 0$, $x = -1$, and $x = 1$

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

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Isolate the logarithmic term

Convert to exponential form and solve

Verify the solution

Answer: