what is the true solution to the logarithmic equation?

$\log_2\log_2(\sqrt{4x}) = 1$

$x = -4$
$x = 0$
$x = 2$
$x = 4$

Question

what is the true solution to the logarithmic equation?

$\log_2\log_2(\sqrt{4x}) = 1$

$x = -4$
$x = 0$
$x = 2$
$x = 4$

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" ], "new_concepts": [], "current_concepts": [ "Logarithmic Equations", "Logarithmic Functions" ] } ## Apply the outer logarithm definition $$ \log_2[\log_2(\sqrt{4x})] = 1 \implies \log_2(\sqrt{4x}) = 2^1 = 2 $$ ## Apply the inner logarithm definition $$ \log_2(\sqrt{4x}) = 2 \implies \sqrt{4x} = 2^2 = 4 $$ ## Solve for x and verify $$ \begin{aligned} 4x &= 4^2 \ 4x &= 16 \ x &= 4 \end{aligned} $$ Checking $x = 4$ in the original equation: $$ \log_2[\log_2(\sqrt{16})] = \log_2[\log_2(4)] = \log_2(2) = 1 $$ The solution is valid.

$x = -4$

$x = 0$

$x = 2$

$x = 4$

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

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QUESTION IMAGE

Question

Explanation:

Apply the outer logarithm definition

Apply the inner logarithm definition

Solve for x and verify

Answer: