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Logarithmic Inequalities

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

Same Base

Solve the logarithmic inequality \[\log_3(x-9)+\log_3(x-7)<1.\]

How many integer solutions does the following inequality have: \[\log_{3}x^{9}+\left( \log_{3}x \right)^2 \le 10?\]

If the solution to the inequality \[ (\log_{5} x)^2 < 5 - \log_{5} x^{4} \] is \( a < x < b \), what is the value of \( \frac{1}{ab} \)?

Solve the logarithmic inequality \[\log_2 (15x-17) \ge \log_2 (x+16).\]

How many positive integers \(x\) satisfy the inequality \( \log_{0.1} (x-15) - \log_{0.1} (301-x) > 0 \)?

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