Question:
The minimum value of \( \frac{x}{\log x} \) is
The minimum value of \( \frac{x}{\log x} \) is
Show Hint
For expressions involving \( \frac{x}{\log x} \), minimum usually occurs at \( x = e \).
Updated On: Apr 23, 2026
- \( e \)
- \( \frac{1}{e} \)
- \( e^2 \)
- \( e^3 \)
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The Correct Option is A
Solution and Explanation
Concept:
Use differentiation to find minimum value.
Step 1: Let \( f(x) = \frac{x}{\log x} \). \[ f'(x) = \frac{\log x \cdot 1 - x \cdot \frac{1}{x}}{(\log x)^2} = \frac{\log x - 1}{(\log x)^2} \]
Step 2: Set derivative to zero: \[ \log x - 1 = 0 \Rightarrow \log x = 1 \Rightarrow x = e \]
Step 3: Minimum value: \[ f(e) = \frac{e}{1} = e \] Final Answer: \[ e \]
Step 1: Let \( f(x) = \frac{x}{\log x} \). \[ f'(x) = \frac{\log x \cdot 1 - x \cdot \frac{1}{x}}{(\log x)^2} = \frac{\log x - 1}{(\log x)^2} \]
Step 2: Set derivative to zero: \[ \log x - 1 = 0 \Rightarrow \log x = 1 \Rightarrow x = e \]
Step 3: Minimum value: \[ f(e) = \frac{e}{1} = e \] Final Answer: \[ e \]
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