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# Logarithmic Inequalities

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

\[\log_{4} x^2\ + \log_{x} 16\ - 5 < 0\]

How many positive integer solutions can satisfy the inequality above?

\[\log _{ 3 }{ \frac { \left| { x }^{ 2 }-4x \right| +3 }{ { x }^{ 2 }+\left| x-5 \right| } } \ge 0 \]

Solve the following for \( x \).

\[\log_{5n} 30\sqrt{5} \ge \log_{4n} 48\]

Over the domain \( n > 1 \), let \(M\) be the smallest value of \(n\) that satisfies the above inequality.

What is \( M^3 \)?

Given that

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

determine \( \lfloor \log_x 7^{100} \rfloor \).

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

find the minimum positive value that \(x\) can have that satisfies the inequality above

write your answer as the square of the number

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