Question

Loudness of Sound Problems 127–130 use the following discussion: The loudness $L\left(x\right),$measured in decibels (dB), of a sound of intensity $x,$measured in watts per square meter, is defined as role="math" localid="1647162539922" $L\left(x\right)=10\log \frac{x}{I_0},$where $I_0={10}^{-12}$watt per square meter is the least intense sound that a human ear can detect. Determine the loudness, in decibels, of each of the following sounds?

Normal conversation: intensity of $x={10}^{-7}$watt per square meter.

Short answer

The loudness of normal conversation is$50$decibel.

Step-by-step solution

Step 1. Given information

It is given that $L\left(x\right)=10\log \frac{x}{I_0},$where $I_0={10}^{-12}$watt per square. We need to determine the loudness in decibels of normal conversation intensity of$x={10}^{-7}$watt per second.

Step 2. Simplify

Substitute ${10}^{-7}$to $x$and ${10}^{-12}$to $I$in,

$L\left(x\right)=10\log \frac{x}{I_0}.$

$L\left({10}^{-7}\right)=10\log \left(\frac{{10}^{-7}}{{10}^{-12}}\right).$

$=10\log \left({10}^{-7+12}\right).$

$=10\log \left({10}^5\right).$

$=10\left(5\right).$

$=50.$

$L\left({10}^{-7}\right)=50$decibel.