what is the true solution to $\ln 20 + \ln 5 = 2 \ln x$?

$x = 5$
$x = 10$
$x = 50$
$x = 100$

Question

what is the true solution to $\ln 20 + \ln 5 = 2 \ln x$?

$x = 5$
$x = 10$
$x = 50$
$x = 100$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Logarithmic Properties", "Logarithmic Equations" ], "new_concepts": [], "current_concepts": [ "Logarithmic Properties", "Logarithmic Equations" ] } ## Apply logarithmic properties to simplify both sides $$ \begin{aligned} \ln(20) + \ln(5) &= \ln(20 \cdot 5) = \ln(100) \ 2\ln(x) &= \ln(x^2) \quad (\text{for } x > 0) \end{aligned} $$ ## Equate the arguments and solve for x $$ \begin{aligned} \ln(100) &= \ln(x^2) \ 100 &= x^2 \ x &= \pm 10 \end{aligned} $$ ## Verify the domain of the original equation $$ \begin{aligned} &\text{Since } \ln(x) \text{ requires } x > 0: \ &x = 10 \quad (\text{valid}) \ &x = -10 \quad (\text{extraneous}) \end{aligned} $$

$x = 5$

$x = 10$

$x = 50$

$x = 100$

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

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Apply logarithmic properties to simplify both sides

Equate the arguments and solve for x

Verify the domain of the original equation

Answer: