Question

which of the following is true regarding the solution to the logarithmic equation below?

\\(\log_{2}(x+11) = 4\\)

\\(x+11 = 2^{4}\\)

\\(x+11 = 16\\)

\\(x = 5\\)

• \\(x = 5\\) is not a true solution because \\(\log_{5}(16) \

eq 2\\)

• \\(x = 5\\) is not a true solution because \\(\log_{5}(16) \

eq 4\\)

• \\(x = 5\\) is a true solution because \\(\log_{2}(16) = 4\\)
• \\(x = 5\\) is a true solution because \\(\log_{4}(16) = 2\\)

Explanation:

<pre_analysis>
{
"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"
]
}
</pre_analysis>

<reasoning>

Solve the logarithmic equation

\[

$$\begin{aligned} \log_2(x+11) &= 4 \\ x+11 &= 2^4 \\ x+11 &= 16 \\ x &= 5 \end{aligned}$$

\]

Verify the solution

\[

$$\begin{aligned} \text{Substitute } x = 5 \text{ back into the original equation:} \\ \log_2(5+11) &= \log_2(16) \\ \log_2(16) &= 4 \quad (\text{since } 2^4 = 16) \end{aligned}$$

\]

Match with the given options

\[

$$\begin{aligned} &\text{The statement } "x = 5 \text{ is a true solution because } \log_2(16) = 4" \text{ is correct.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(x = 5\) is not a true solution because \(\log_5(16)
eq 2\)</mcq-option>
<mcq-option>(B) \(x = 5\) is not a true solution because \(\log_5(16)
eq 4\)</mcq-option>
<mcq-correct>(C) \(x = 5\) is a true solution because \(\log_2(16) = 4\)</mcq-correct>
<mcq-option>(D) \(x = 5\) is a true solution because \(\log_4(16) = 2\)</mcq-option>
</answer>

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

Answer:

<pre_analysis>
{
"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"
]
}
</pre_analysis>

<reasoning>

Solve the logarithmic equation

\[

$$\begin{aligned} \log_2(x+11) &= 4 \\ x+11 &= 2^4 \\ x+11 &= 16 \\ x &= 5 \end{aligned}$$

\]

Verify the solution

\[

$$\begin{aligned} \text{Substitute } x = 5 \text{ back into the original equation:} \\ \log_2(5+11) &= \log_2(16) \\ \log_2(16) &= 4 \quad (\text{since } 2^4 = 16) \end{aligned}$$

\]

Match with the given options

\[

$$\begin{aligned} &\text{The statement } "x = 5 \text{ is a true solution because } \log_2(16) = 4" \text{ is correct.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(x = 5\) is not a true solution because \(\log_5(16)
eq 2\)</mcq-option>
<mcq-option>(B) \(x = 5\) is not a true solution because \(\log_5(16)
eq 4\)</mcq-option>
<mcq-correct>(C) \(x = 5\) is a true solution because \(\log_2(16) = 4\)</mcq-correct>
<mcq-option>(D) \(x = 5\) is a true solution because \(\log_4(16) = 2\)</mcq-option>
</answer>

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