select all of the potential solution(s) of the equation $2\log_{5}x = \log_{5}4$.

$x = -2$
$x = 4$
$x = 16$
$x = -10$
$x = 2$

Question

select all of the potential solution(s) of the equation $2\log_{5}x = \log_{5}4$.

$x = -2$
  $x = 4$
  $x = 16$
  $x = -10$
  $x = 2$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Logarithmic Power Rule", "Solving Logarithmic Equations" ], "new_concepts": [], "current_concepts": [ "Logarithmic Power Rule", "Solving Logarithmic Equations" ] } ## Apply the power rule of logarithms Using the *Logarithmic Power Rule* knowledge point $$ \begin{aligned} 2\log_{5}x &= \log_{5}(x^2) \end{aligned} $$ ## Equate the arguments Using the *Solving Logarithmic Equations* knowledge point $$ \begin{aligned} \log_{5}(x^2) &= \log_{5}4 \ x^2 &= 4 \end{aligned} $$ ## Solve for x Using the *Solving Logarithmic Equations* knowledge point $$ \begin{aligned} x &= \pm\sqrt{4} \ x &= 2 \quad \text{or} \quad x = -2 \end{aligned} $$ ## Check for extraneous solutions Using the *Solving Logarithmic Equations* knowledge point $$ \begin{aligned} &\text{For } x = 2: \quad 2\log_{5}(2) \text{ is defined since } 2 > 0. \ &\text{For } x = -2: \quad 2\log_{5}(-2) \text{ is undefined since } -2 \le 0. \ &\text{Thus, the only valid solution is } x = 2. \end{aligned} $$

$x = -2$

$x = 4$

$x = 16$

$x = -10$

$x = 2$

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

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QUESTION IMAGE

Question

Explanation:

Apply the power rule of logarithms

Equate the arguments

Solve for x

Check for extraneous solutions

Answer: