LOG10E.CONST

Application:

Converting Sound Intensity to Decibels

The decibel (dB) scale is a logarithmic scale used to measure sound intensity. The formula for sound level in decibels is:

$L_{dB}=10\cdot {\log }_{10}\left(\frac{I}{I_0}\right)$

where I is the sound intensity being measured, and I0​ is the reference intensity (the threshold of human hearing, approximately 10−12 W/m2).

Imagine you are a scientist modeling the exponential decay of a sound wave's intensity as it travels through a medium. The intensity of the sound wave can be described by the formula:

$I(d)=I_{initial}\cdot e^{-kd}$

where d is the distance from the source, and k is the decay constant of the medium.

To see how the sound level in decibels changes with distance, you would need to substitute the intensity function into the decibel formula:

$L_{dB}(d)=10\cdot {\log }_{10}\left(\frac{I_{initial}\cdot e^{-kd}}{I_0}\right)$

Using the properties of logarithms, you can simplify this expression.

$L_{dB}(d)=10\cdot \left[{\log }_{10}\left(\frac{I_{initial}}{I_0}\right)+{\log }_{10}(e^{-kd})\right]$

$L_{dB}(d)=10\cdot \left[{\log }_{10}\left(\frac{I_{initial}}{I_0}\right)-kd\cdot {\log }_{10}(e)\right]$

This is where the LOG10E.CONST constant comes in. The value of log10​(e) is approximately 0.434294.

Let's say you have an initial sound intensity Iinitial​=10−6 W/m2 and a decay constant k=0.25 m−1. You want to calculate the sound level in decibels at various distances.

Using the simplified formula:

$L_{dB}(d)=10\cdot \left[{\log }_{10}\left(\frac{10^{-6}}{10^{-12}}\right)-0.25d\cdot \text{LOG10E.CONST}\right]$

$L_{dB}(d)=10\cdot \left[{\log }_{10}(10^6)-0.25d\cdot 0.434294\right]$

$L_{dB}(d)=10\cdot \left[6-0.1085735d\right]$

$L_{dB}(d)=60-1.085735d$

You can now use this simplified linear equation to quickly calculate the decibel level at different distances, as shown in the table below. The LOG10E.CONST value was crucial for converting the natural exponential decay into a linear relationship on the base-10 decibel scale.