The loudness, [tex]$L$[/tex], measured in decibels (Db), of a sound intensity, [tex]$l$[/tex], measured in watts per square meter, is defined as [tex]$L=10 \log \frac{l}{l_0}$[/tex], where [tex]$l_0=10^{-12}$[/tex] and is the least intense sound a human ear can hear. What is the approximate loudness of a rock concert with a sound intensity of [tex]$10^{-1}$[/tex]?
Answer (1)
Substitute the given values l = 1 0 − 1 and l 0 = 1 0 − 12 into the formula L = 10 lo g 10 l 0 l .
Simplify the fraction: 1 0 − 12 1 0 − 1 = 1 0 11 .
Apply the logarithm: L = 10 lo g 10 ( 1 0 11 ) = 10 × 11 .
Calculate the loudness: L = 110 dB. The final answer is 110
Explanation
Understanding the Problem We are given the formula for loudness L in decibels (dB) as: L = 10 lo g 10 l 0 l where l is the sound intensity in watts per square meter, and l 0 = 1 0 − 12 watts per square meter is the reference intensity (the least intense sound a human ear can hear). We are asked to find the loudness of a rock concert with a sound intensity of l = 1 0 − 1 watts per square meter.
Substituting Values We substitute the given values into the formula: L = 10 lo g 10 1 0 − 12 1 0 − 1 Now we simplify the fraction inside the logarithm: 1 0 − 12 1 0 − 1 = 1 0 − 1 − ( − 12 ) = 1 0 − 1 + 12 = 1 0 11 So we have: L = 10 lo g 10 ( 1 0 11 )
Calculating Loudness Using the property of logarithms that lo g 10 ( 1 0 x ) = x , we have: L = 10 × 11 = 110 Therefore, the loudness of the rock concert is approximately 110 dB.
Final Answer The approximate loudness of a rock concert with a sound intensity of 1 0 − 1 watts per square meter is 110 dB.
Examples
The decibel scale is used in many real-world applications, such as measuring the loudness of music, the intensity of noise pollution, and the sensitivity of microphones. For example, understanding decibels helps audio engineers design sound systems, helps city planners manage noise levels, and helps doctors diagnose hearing problems. The formula L = 10 lo g 10 l 0 l allows us to quantify and compare the loudness of different sounds, which is crucial in many fields.
