Problem
The loudness level of a sound can be expressed by comparing the sound's intensity to the intensity of a sound barely audible to the human ear. The formula $\mathrm{D}=10\left(\log \mathrm{I}-\log \mathrm{I}_{0}\right)$ describes the loudness level of a sound, $\mathrm{D}$, in decibels, where $\mathrm{I}$ is the intensity of the sound, in watts per meter ${ }^{2}$, and $\mathrm{I}_{0}$ is the intensity of a sound barely audible to the human ear. Use this information to answer parts (a) and (b) below. a. Express the formula so that the expression in parentheses is written as a single logarithm.
Solution
Step 1 ：Given the formula $D=10(\log I-\log I_0)$, we can use the properties of logarithms to combine the two logarithms into one. The property $\log a - \log b = \log \left(\frac{a}{b}\right)$ allows us to rewrite the formula as $D = 10 \log \left(\frac{I}{I_0}\right)$
Step 2 ：So, the formula expressed as a single logarithm is $D = 10 \log \left(\frac{I}{I_0}\right)$
Step 3 ：\(\boxed{D = 10 \log \left(\frac{I}{I_0}\right)}\)
