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# Logarithmic Inequalities

Logarithmic inequalities are useful in analyzing situations involving repeated multiplication, such as interest and exponential decay.

\[\large \log_{0.3}{(x-1)}<\log_{0.09}{(x-1)}\]

Find the range of \(x\) that satisfies the inequality above.

Find the minimum value of the positive integer \(x\) satisfying the inequality \[\frac{\log_{10} (2x-1)}{2}+\log_{10} \sqrt{x-9} \geq 1. \]

*Source: RMO*

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