Exponential and Logarithmic Functions - Applications of Exponential and Logarithmic Functions

Loudness is a subjective measure of how strong a sound feels, as it can vary between individuals. A more objective way to measure sound strength is the sound intensity level, \(L\), given by the formula \(L=10\lg \frac{I}{{{I}_{0}}}\),

where \(I\) represents the sound intensity in W/m\(^{2}\), and \({{I}_{0}}\) is the minimum sound intensity detectable by the human ear (also measured in W/m\(^{2}\)). The unit of \(L\) is decibels (dB).

1. What is the lowest value of \(L\) (in dB) that the human ear can detect?
2. If \(L=90\) dB when \(I={{10}^{-3}}\) W/m\(^{2}\), find the minimum sound intensity detectable by the human ear.
3. Find the sound intensity level for a refrigerator with a sound intensity of \(4\times {{10}^{-8}}\) W/m\(^{2}\).
4. Given that the sound intensity level of a machine is \(120\) dB, find the corresponding sound intensity \(I\).