which of the following is true regarding the solutions to the logarithmic equation below?

$2\log_{6}(x) = 2$
$\log_{6}(x^2) = 2$
$x^2 = 6^2$
$x^2 = 36$
$x = 6, -6$

* $x = 6$ and $x = -6$ are true solutions
* $x = 6$ and $x = -6$ are extraneous solutions
* $x = 6$ is a true solution and $x = -6$ is an extraneous solution
* $x = 6$ is an extraneous solution and $x = -6$ is a true solution

Question

which of the following is true regarding the solutions to the logarithmic equation below?

$2\log_{6}(x) = 2$
$\log_{6}(x^2) = 2$
$x^2 = 6^2$
$x^2 = 36$
$x = 6, -6$

* $x = 6$ and $x = -6$ are true solutions
* $x = 6$ and $x = -6$ are extraneous solutions
* $x = 6$ is a true solution and $x = -6$ is an extraneous solution
* $x = 6$ is an extraneous solution and $x = -6$ is a true solution

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" ] } ## Identify the original equation and its domain restriction $$ \begin{aligned} &2\log_{6}(x) = 2 \ &\text{Domain restriction: } x > 0 \end{aligned} $$ ## Evaluate the proposed solutions against the domain $$ \begin{aligned} &x = 6 \implies 6 > 0 \quad (\text{True solution}) \ &x = -6 \implies -6 gtr 0 \quad (\text{Extraneous solution}) \end{aligned} $$ ## Match with the correct multiple-choice option $$ x = 6 \text{ is a true solution and } x = -6 \text{ is an extraneous solution} $$

x = 6 and x = -6 are true solutions

x = 6 and x = -6 are extraneous solutions

x = 6 is a true solution and x = -6 is an extraneous solution

x = 6 is an extraneous solution and x = -6 is a true solution

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

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QUESTION IMAGE

Question

Explanation:

Identify the original equation and its domain restriction

Evaluate the proposed solutions against the domain

Match with the correct multiple-choice option

Answer: