what is the true solution to the equation below?

$\ln e^{\ln x} + \ln e^{\ln x^2} = 2 \ln 8$

* $x = 2$
* $x = 4$
* $x = 8$
* $x = 64$

Question

what is the true solution to the equation below?

$\ln e^{\ln x} + \ln e^{\ln x^2} = 2 \ln 8$

* $x = 2$
* $x = 4$
* $x = 8$
* $x = 64$

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": [ "Inverse Properties of Logarithms", "Solving Logarithmic Equations" ], "new_concepts": [], "current_concepts": [ "Inverse Properties of Logarithms", "Solving Logarithmic Equations", "Logarithmic Power Rule" ] } ## Simplify the logarithmic terms Using the *Inverse Properties of Logarithms* knowledge point $$ \begin{aligned} &\ln(e^{\ln(x)}) = \ln(x) \ &\ln(e^{\ln(x^2)}) = \ln(x^2) \end{aligned} $$ ## Rewrite the equation Using the *Logarithmic Power Rule* knowledge point $$ \begin{aligned} &\ln(x) + \ln(x^2) = 2\ln(8) \ &\ln(x) + 2\ln(x) = 2\ln(8) \ &3\ln(x) = 2\ln(8) \end{aligned} $$ ## Solve for x Using the *Solving Logarithmic Equations* knowledge point $$ \begin{aligned} &3\ln(x) = 2\ln(2^3) \ &3\ln(x) = 6\ln(2) \ &\ln(x) = 2\ln(2) \ &\ln(x) = \ln(2^2) \ &x = 4 \end{aligned} $$ ## Verify the solution Using the *Solving Logarithmic Equations* knowledge point $$ \begin{aligned} &\text{For } x = 4 > 0 \text{, the arguments of the logarithms are positive and valid.} \end{aligned} $$

$x = 2$

$x = 4$

$x = 8$

$x = 64$

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

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QUESTION IMAGE

Question

Explanation:

Simplify the logarithmic terms

Rewrite the equation

Solve for x

Verify the solution

Answer: