Question

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

\\(x = 5\\)
\\(x = 10\\)
\\(x = 50\\)
\\(x = 100\\)

Explanation:

<pre_analysis>
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<reasoning>

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

\[

$$\begin{aligned} &\text{Since } \ln(x) \text{ requires } x > 0: \\ &x = 10 \quad (\text{valid}) \\ &x = -10 \quad (\text{extraneous}) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>\(x = 5\)</mcq-option>
<mcq-correct>\(x = 10\)</mcq-correct>
<mcq-option>\(x = 50\)</mcq-option>
<mcq-option>\(x = 100\)</mcq-option>
</answer>

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"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Logarithmic Equations"
]
}
</post_analysis>

Answer:

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</pre_analysis>

<reasoning>

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

\[

$$\begin{aligned} &\text{Since } \ln(x) \text{ requires } x > 0: \\ &x = 10 \quad (\text{valid}) \\ &x = -10 \quad (\text{extraneous}) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>\(x = 5\)</mcq-option>
<mcq-correct>\(x = 10\)</mcq-correct>
<mcq-option>\(x = 50\)</mcq-option>
<mcq-option>\(x = 100\)</mcq-option>
</answer>

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