Algebra

Logarithmic Inequalities

Logarithmic Inequalities: Level 4 Challenges

         

log4x2 +logx16 5<0\log_{4} x^2\ + \log_{x} 16\ - 5 < 0

How many positive integer solutions can satisfy the inequality above?

log3x24x+3x2+x50\log _{ 3 }{ \frac { \left| { x }^{ 2 }-4x \right| +3 }{ { x }^{ 2 }+\left| x-5 \right| } } \ge 0

Solve the following for x x .

Try Fun with inequalities-1 and Fun with inequalities-2

log5n305log4n48\log_{5n} 30\sqrt{5} \ge \log_{4n} 48

Over the domain n>1 n > 1 , let MM be the smallest value of nn that satisfies the above inequality.

What is M3? M^3?

Given that

0.161<logx2<0.162,0.256<logx3<0.257,0.375<logx5<0.376, \begin{array} {lll} 0.161 & < \log_x 2 < & 0.162, \\ 0.256 & < \log_x 3 < & 0.257, \\ 0.375 & < \log_x 5 < & 0.376,\\ \end{array}

determine logx7100 \lfloor \log_x 7^{100} \rfloor .

ln(x2+2x+1)ln(x4+1)\ln { (x^{ 2 }+\sqrt { 2 } x+1) } \le \ln { (x^{ 4 }+1) }

find the minimum positive value that xx can have that satisfies the inequality above

write your answer as the square of the number

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