Prove that the logarithmic function is strictly increasing on $ (0, \infty) $.

Q: Prove that the logarithmic function is strictly increasing on $ (0, \infty) $.

Explanation: Let $ f(x) = \log x $. Then, \[ f'(x) = \frac{d}{dx}(\log x) = \frac{1}{x} \] For $ x > 0 $, we have: \[ f'(x) = \frac{1}{x} > 0 \] Since the derivative is positive for all $ x > 0 $, the function $ f(x) = \log x $ is strictly increasing on $ (0, \infty) $.

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Explanation: Let $ f(x) = \log x $.

Then,  
$$
f'(x) = \frac{d}{dx}(\log x) = \frac{1}{x}
$$

For $ x > 0 $, we have:  
$$
f'(x) = \frac{1}{x} > 0
$$

Since the derivative is positive for all $ x > 0 $,  
the function $ f(x) = \log x $ is strictly increasing on $ (0, \infty) $.

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