Logarithmic Inequalities

Logarithmic Inequalities: Level 3 Challenges


If the domain of the solutions of the inequality (log2x)2<2log2x+3(\log_{2}{x})^2 < 2\log_{2}{x}+3 is from aa to bb (both exclusive). Find the value of a+ba+b.

log0.3(x1)<log0.09(x1)\large \log_{0.3}{(x-1)}<\log_{0.09}{(x-1)}

Find the range of xx that satisfies the inequality above.

Find the minimum value of the positive integer xx satisfying the inequality log10(2x1)2+log10x91.\frac{\log_{10} (2x-1)}{2}+\log_{10} \sqrt{x-9} \geq 1.

Source: RMO

Find the smallest positive integer kk such that 1log3k2015!+1log4k2015!++1log2015k2015!>2015\dfrac{1}{\log_{3^k}2015!}+\dfrac{1}{\log_{4^k}2015!}+\ldots+\dfrac{1}{\log_{2015^k}2015!}>2015

logx(2x)+log2(x)3\large \log_{x}(2x)+\log_{2}(x)\geq 3 The inequality above has a solution in the form (a,+)(a,+\infty). Find the value of aa


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