The intensity, or loudness, of a sound can be measured in decibels $(d B)$, according to the equation $/(d B)=10 olog \left[\frac{1}{l_0}\right]$, where $I$ is the intensity of a given sound and $I_0$ is the threshold of hearing intensity. What is the intensity, in decibels, $[I(d B)]$, when $I=10^8\left(I_0\right)$ ?
Answer (2)
Substitute I = 1 0 8 I 0 into the decibel formula: I ( d B ) = 10 lo g [ I 0 I ] .
Simplify the expression: I ( d B ) = 10 lo g [ I 0 1 0 8 I 0 ] = 10 lo g ( 1 0 8 ) .
Apply the logarithm property: 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 [ I 0 I ] , where I is the intensity of the sound and I 0 is the threshold of hearing intensity. We are also given that I = 1 0 8 I 0 .
Substituting the Value of I We need to find the intensity in decibels, I ( d B ) , when I = 1 0 8 I 0 . We substitute the given value of I into the formula:
Plugging in the values I ( d B ) = 10 lo g [ I 0 1 0 8 I 0 ]
Simplifying the Expression We can simplify the expression by canceling out I 0 :
Further Simplification I ( d B ) = 10 lo g ( 1 0 8 )
Applying Logarithm Properties Using the property of logarithms that lo g ( 1 0 x ) = x , we have:
Calculating the Final Value I ( d B ) = 10 ∗ 8 = 80
Final Answer Therefore, the intensity in decibels when I = 1 0 8 I 0 is 80 dB.
Examples
Understanding decibels is crucial in many real-world scenarios. For instance, sound engineers use decibel measurements to optimize audio equipment and ensure safe listening levels at concerts. In urban planning, decibel levels help assess noise pollution from traffic and industrial activities, guiding the implementation of noise reduction strategies. Moreover, audiologists rely on decibel measurements to diagnose hearing loss and prescribe appropriate hearing aids, improving the quality of life for individuals with auditory impairments. The formula I ( d B ) = 10 lo g [ I 0 I ] is a cornerstone in these applications, bridging the gap between physical sound intensity and human perception of loudness.
When the intensity of sound is I = 1 0 8 I 0 , we can calculate the intensity in decibels using the formula I ( d B ) = 10 lo g ( I 0 I ) . After substituting and simplifying, we find that the intensity is 80 d B .
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