Question

the loudness, l, measured in decibels (db), of a sound intensity, i, measured in watts per square meter, is defined as $l = 10\log\frac{i}{i_0}$, where $i_0 = 10^{-12}$ and is the least intense sound a human ear can hear. what is the approximate loudness of a dinner conversation with a sound intensity of $10^{-7}$?
-58 db
-50 db
9 db
50 db

Explanation:

Step1: Substitute values into formula

Given $L = 10\log\frac{I}{I_0}$, with $I = 10^{-7}$ and $I_0=10^{-12}$. Then $\frac{I}{I_0}=\frac{10^{-7}}{10^{-12}}$.
Using the rule $\frac{a^m}{a^n}=a^{m - n}$, we have $\frac{10^{-7}}{10^{-12}}=10^{-7-(-12)} = 10^{5}$.

Step2: Calculate the logarithm

$L = 10\log(10^{5})$.
Since $\log(a^b)=b\log(a)$ and $\log(10) = 1$, then $\log(10^{5})=5\log(10)=5$.

Step3: Find the loudness

$L = 10\times5=50$.

Answer:

D. 50 Db