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Logarithmic Inequalities

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

Level 3

         

If the domain of the solutions of the inequality \((\log_{2}{x})^2 < 2\log_{2}{x}+3\) is from \(a\) to \(b\) (both exclusive). Find the value of \(a+b\).

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

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

Find the smallest positive integer \(k\) such that \[\dfrac{1}{\log_{3^k}2015!}+\dfrac{1}{\log_{4^k}2015!}+\ldots+\dfrac{1}{\log_{2015^k}2015!}>2015\]

\[\large \log_{x}(2x)+\log_{2}(x)\geq 3\] The inequality above has a solution in the form \((a,+\infty)\). Find the value of \(a\)

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