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A quiet rustle of leaves has a power of about 0.00000000001Β W/m20.00000000001\text{ W/m}^2, while a rock concert blasts sound at 1Β W/m21\text{ W/m}^2. That is a difference of one hundred billion times, yet your ears can listen to both without instantly tearing apart.
How do we compress a scale that spans over twelve orders of magnitude into numbers that actually make intuitive sense?

Sound Intensity and the Human Ear

Sound is energy travelling through air as pressure waves. We measure sound intensity (II) as the power passing through a given area, with standard units of watts per square meter (W/m2\text{W/m}^2).
The quietest sound a healthy human ear can detect at a frequency of 1Β kHz1\text{ kHz} is called the threshold of hearing, labeled I0I_0, which equals exactly 10βˆ’12Β W/m210^{-12}\text{ W/m}^2.
Because physical intensity multiplies by powers of 1010 while our perception grows in steady steps, linear units like watts fail to capture how loud something feels. So how do we translate raw power into perceived volume?

The Decibel Formula

In the 1920s, engineers at Bell Telephone Laboratories created the 'Bel' to quantify signal loss across telephone cables, naming it after telephone pioneer Alexander Graham Bell. Because a Bel was too broad for everyday use, scientists adopted the decibel (dB), which is one-tenth of a Bel.
A logarithm base 10, written log⁑10(x)\log_{10}(x), answers the question: 'What power must we raise 1010 to in order to get xx?' By comparing any sound intensity II directly to the baseline I0I_0, the decibel level β\beta measures the ratio on an exponential scale of 10.
Now that we have the formula β=10log⁑10(II0)\beta = 10\log_{10}\left(\frac{I}{I_0}\right), how do we use it to calculate the decibel rating of a specific, real-world noise?

Calculating Decibels from Sound Intensity