Waste less time on Facebook — follow Brilliant.
×

Logarithmic Inequalities

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

Multiple Terms

         

If the range of \(x\) that satisfies \[{5}^{\log_{9}x} \cdot {x}^{\log_{9}{5}}-3({5}^{\log_{9}x}+{x}^{\log_{9}{5}})+5 < 0\] is \(\alpha < x < \beta,\) what is the value of \(\alpha+\beta?\)

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

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

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

What is the number of integers \(x\) that satisfy \[\log_{\frac{1}{2}} \frac{4}{x} \cdot \log_{\frac{1}{2}} \frac{x}{8} \geq -2 ?\]

×

Problem Loading...

Note Loading...

Set Loading...