Logarithmic Inequalities
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 \(\alpha\beta ?\)