Algebra
# Logarithmic Inequalities

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.

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