Question

the loudness, l, measured in decibels (db), of a sound intensity, i, measured in watts per square meter, is defined as l = 10log\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 $L = 10\log\frac{I}{I_0}$, $I = 10^{-7}$ and $I_0=10^{-12}$. Substitute these values: $L = 10\log\frac{10^{-7}}{10^{-12}}$.

Step2: Simplify the fraction inside the logarithm

Using the rule $\frac{a^m}{a^n}=a^{m - n}$, we have $\frac{10^{-7}}{10^{-12}}=10^{-7-(-12)} = 10^5$. So $L = 10\log(10^5)$.

Step3: Evaluate the logarithm

Since $\log_{10}(10^x)=x$, then $\log(10^5) = 5$. So $L=10\times5$.

Step4: Calculate the final value

$L = 50$.

Answer:

50 Db