Prove \( \int_{a}^{b} \frac{\log x}{x} d x=\frac{1}{2} \log \left(\frac{b}{a}\right) \log (a b) \).

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Prove \( \int_{a}^{b} \frac{\log x}{x} d x=\frac{1}{2} \log \left(\frac{b}{a}\right) \log (a b) \).
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