Algebra
# Logarithmic Inequalities

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

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*