Problem
Express the formula so that the expression in parentheses is written as a single logarithm: 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 $D=10\left(\log I-\log I_{0}\right)$ describes the loudness level of a sound, $D$, in decibels, where $I$ 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. \[ D=\square \]
Solution
Step 1 ：Given the formula for the loudness level of a sound, $D=10\left(\log I-\log I_{0}\right)$, where $I$ 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.
Step 2 ：Using the properties of logarithms, we can combine the two logarithms in the parentheses into a single logarithm by division. This is because the difference of two logarithms is equal to the logarithm of the quotient of their arguments.
Step 3 ：So, we can rewrite the formula as $D=10\log \left(\frac{I}{I_{0}}\right)$.
Step 4 ：\(\boxed{D=10\log \left(\frac{I}{I_{0}}\right)}\) is the final answer.
