Chapter 3, Problem 38RE

Recall from Section $1.8$ that the loudness $\beta$ of a sound in decibels (dB) is given by $\beta =10\log \left(I/I_0\right)$ , where $I$ is the intensity of the sound in watts per square meter $\left({\text{W/m}}^{\text{2}}\right)$ and $I_0$ is a constant that is approximately the intensity of a sound at the threshold of human hearing. Find the rate of change of $\beta$ with respect to $I$ at the point where

(a) $I/I_0=10$

(b) $I/I_0=100$

(c) $I/I_0=1000$ .