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Logarithmic Inequalities

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

Problem Solving


What is the range of \(x\) that satisfies the above logarithmic inequality?

The sound intensity level (measured in decibels) is given by the formula: \[L_I=10\log\left(\frac{I}{I_0}\right)\text{ (dB)},\] where \(I\) denotes sound intensity and \(I_0\) is the reference sound intensity.

An individual is diagnosed as moderate hearing loss when his/her threshold of sound perception is greater than 40 dB and less than or equal to 55 dB. Given that \(I_0=1\text{ pW/m}^2,\) what is the range of the threshold of sound intensity \(x\) that a person with moderate hearing loss can perceive?

Details and Assumptions:
- The sound intensity level of a normal conversation is about 50~60 dB, and that of a quiet library is about 40 dB.
- Assume that \(\sqrt{10}\approx3.\)

\(A, B \) and \(C\) are positive integers satisfying \( A \geq 2, B \geq 4, C \geq 8 \). What is the minimum integer value of

\[ \log_A (BC) + \log_B (CA) + \log_C (AB) ? \]

Details and assumptions

The minimum integer value of a set, is the smallest integer that appears in the set. As an explicit example, the minimum integer value of the set of numbers \(x\) satisfying \( x \geq 12.3\) is 13.

If one root of the following quadratic equation \[x^2-2x\log_{9}a+4\log_{9}a=0 \] is positive and the other root is negative, what is the range of \(a?\)

If the range of \(x\) that satisfies the inequality \[{4}^4 {7}^4 x^3 > x^{\log_{28}x}\] is \(\alpha < x < \beta,\) what is the value of \(\alpha\beta ?\)


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