Algebra

Logarithmic Inequalities

Logarithmic Inequalities - Multiple Terms

         

If the range of xx that satisfies 5log9xxlog953(5log9x+xlog95)+5<0{5}^{\log_{9}x} \cdot {x}^{\log_{9}{5}}-3({5}^{\log_{9}x}+{x}^{\log_{9}{5}})+5 < 0 is α<x<β,\alpha < x < \beta, what is the value of α+β?\alpha+\beta?

What is the range of xx that satisfies the logarithmic inequality (log4x)25log4x+60?(\log_{4}x)^2-5\log_{4}x+6 \leq 0 ?

If the range of xx that satisfies the logarithmic inequality (log5x)2log5125x20(\log_{5}x)^2-\log_{5}125x^{2} \leq 0 is αxβ,\alpha \leq x \leq \beta, what is the value of αβ?\alpha\beta?

If the logarithmic inequality (log4x)2log4ax2(\log_{4}x)^2 \geq \log_{4}ax^2 is true for all positive numbers x,x, what is the range of a?a ?

What is the number of integers xx that satisfy log124xlog12x82?\log_{\frac{1}{2}} \frac{4}{x} \cdot \log_{\frac{1}{2}} \frac{x}{8} \geq -2 ?

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