One Problem at a Time

Application of Logarithm (Decibel Level of Sound)

Problem 208: Extended exposure to noise that measures 85 dB can cause permanent damage to the inner ear which will result in hearing loss. What is the decibel level of music coming through ear phones with intensity 10^{-2} watts per square inch?

Solution: The loudness level measured in decibels, of a sound of intensity, I, measured in watts per square inch is

(1)   \begin{equation*}10 \log \left( \frac{I}{10^{-12}}\right). \end{equation*}

Our decibel is then given by

(2)   \begin{align*} 10 \log \left( \frac{I}{10^{-12}}\right) & = 10 \log \left( \frac{10^{-2}}{10^{-12}}\right) \\ & = 10 \log(10^{10}) \\ & = 10 \cdot 10 \\ & = 100. \end{align*}

Thus the decibel level of music coming through earphones is 100 dB.

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