The intensity, or loudness, of a sound can be measured in decibels $(d B)$, according to the equation $I(d B)=10 olog \left[\frac{1}{f_0}\right]$, where $l$ is the intensity of a given sound and $l_0$ is the threshold of hearing intensity. What is the intensity, in decibels, $[/(d B)]$, when $l=10^8\left(l_0\right)$ ?
A. 8
B. 9
C. 19
D. 80
Answer (1)
Substitute l = 1 0 8 l 0 into the formula I ( d B ) = 10 lo g 10 ( l 0 l ) .
Simplify the expression to I ( d B ) = 10 lo g 10 ( 1 0 8 ) .
Use the logarithm property to get I ( d B ) = 10 × 8 .
Calculate the final intensity in decibels: 80 .
Explanation
Understanding the Problem We are given the formula for the intensity of sound in decibels: I ( d B ) = 10 lo g 10 ( l 0 l ) , where l is the intensity of the sound and l 0 is the threshold of hearing intensity. We are also given that l = 1 0 8 l 0 . Our goal is to find the intensity in decibels, I ( d B ) .
Substitution Substitute the given value of l into the formula: I ( d B ) = 10 lo g 10 ( l 0 1 0 8 l 0 )
Simplifying the Expression Simplify the fraction inside the logarithm: I ( d B ) = 10 lo g 10 ( 1 0 8 )
Applying Logarithm Properties Use the property of logarithms that lo g 10 ( 1 0 x ) = x : I ( d B ) = 10 × 8
Final Calculation Calculate the final result: I ( d B ) = 80 Therefore, the intensity in decibels is 80 dB.
Examples
Sound intensity is measured in decibels, which is a logarithmic scale. This scale is useful in many real-world applications, such as measuring the loudness of music, the noise level in a factory, or the sound of a jet engine. For example, if you are designing a concert hall, you need to ensure that the sound intensity is not too high, which could damage people's hearing. Or, if you are designing a quiet office space, you need to minimize the sound intensity from outside noise sources. The decibel scale helps us quantify and manage sound levels in various environments.
