how does the value of $\log_{2} 100$ compare with the value of $\log_{6} 20$?

- the value of $\log_{2} 100$ is about 4 times the value of $\log_{6} 20$
- the value of $\log_{2} 100$ is about $\frac{1}{4}$ times the value of $\log_{6} 20$
- the value of $\log_{2} 100$ is about 3 times the value of $\log_{6} 20$
- the value of $\log_{2} 100$ is about $\frac{1}{3}$ times the value of $\log_{6} 20$

Question

how does the value of $\log_{2} 100$ compare with the value of $\log_{6} 20$?

- the value of $\log_{2} 100$ is about 4 times the value of $\log_{6} 20$
- the value of $\log_{2} 100$ is about $\frac{1}{4}$ times the value of $\log_{6} 20$
- the value of $\log_{2} 100$ is about 3 times the value of $\log_{6} 20$
- the value of $\log_{2} 100$ is about $\frac{1}{3}$ times the value of $\log_{6} 20$

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": [ "Logarithmic Evaluation" ], "new_concepts": [], "current_concepts": [ "Logarithmic Evaluation", "Logarithmic Growth Comparison" ] } ## Estimate the value of the first logarithm Using the *Logarithmic Evaluation* knowledge point $$ \begin{aligned} &2^6 = 64 \ &2^7 = 128 \ &6 < \log_2 100 < 7 \ &\log_2 100 \approx 6.64 \end{aligned} $$ ## Estimate the value of the second logarithm Using the *Logarithmic Evaluation* knowledge point $$ \begin{aligned} &6^1 = 6 \ &6^2 = 36 \ &1 < \log_6 20 < 2 \ &\log_6 20 \approx 1.67 \end{aligned} $$ ## Compare the two estimated values We find the ratio of the first value to the second value: $$ \frac{\log_2 100}{\log_6 20} \approx \frac{6.64}{1.67} \approx 3.98 \approx 4 $$ Thus, the value of $\log_2 100$ is about 4 times the value of $\log_6 20$.

(A) The value of $\log_2 100$ is about 4 times the value of $\log_6 20$

(B) The value of $\log_2 100$ is about $\frac{1}{4}$ times the value of $\log_6 20$

(C) The value of $\log_2 100$ is about 3 times the value of $\log_6 20$

(D) The value of $\log_2 100$ is about $\frac{1}{3}$ times the value of $\log_6 20$

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

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QUESTION IMAGE

Question

Explanation:

Estimate the value of the first logarithm

Estimate the value of the second logarithm

Compare the two estimated values

Answer: