Algebra

Logarithmic Inequalities

Logarithmic Inequalities - Problem Solving

         

What is the range of xx that satisfies the above logarithmic inequality?

The sound intensity level (measured in decibels) is given by the formula: LI=10log(II0) (dB),L_I=10\log\left(\frac{I}{I_0}\right)\text{ (dB)}, where II denotes sound intensity and I0I_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 I0=1 pW/m2,I_0=1\text{ pW/m}^2, what is the range of the threshold of sound intensity xx 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 103.\sqrt{10}\approx3.

A,BA, B and CC are positive integers satisfying A2,B4,C8 A \geq 2, B \geq 4, C \geq 8 . What is the minimum integer value of

logA(BC)+logB(CA)+logC(AB)? \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 xx satisfying x12.3 x \geq 12.3 is 13.

If one root of the following quadratic equation x22xlog9a+4log9a=0x^2-2x\log_{9}a+4\log_{9}a=0 is positive and the other root is negative, what is the range of a?a?

If the range of xx that satisfies the inequality 4474x3>xlog28x{4}^4 {7}^4 x^3 > x^{\log_{28}x} is α<x<β,\alpha < x < \beta, what is the value of αβ?\alpha\beta ?

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