The Decibel Scale — Logarithms Measure Sound
The Decibel Scale
Sound intensity spans an enormous range — a jet engine is about 1 trillion times ($10^{12}$) louder than a whisper. The decibel scale uses logarithms to make this manageable.
$$L = 10 \log_{10}\!\left(\frac{I}{I_0}\right) \text{ dB}$$
where $I$ is the sound intensity and $I_0 = 10^{-12}$ W/m² is the threshold of hearing.
How the Scale Works
- Every +10 dB = 10× the intensity (sounds roughly twice as loud)
- Every +3 dB ≈ 2× the intensity
- 0 dB = threshold of hearing ($I = I_0$)
- 120 dB = threshold of pain ($I = 10^{12} \times I_0$)
| Sound | Intensity (W/m²) | Level (dB) |
|---|---|---|
| Breathing | $10^{-11}$ | 10 |
| Library | $10^{-8}$ | 40 |
| Conversation | $10^{-6}$ | 60 |
| Vacuum cleaner | $10^{-4}$ | 80 |
| Rock concert | $10^{-1}$ | 110 |
| Jet takeoff (30 m) | $10^{2}$ | 140 |
A speaker produces sound at intensity $I = 5 \times 10^{-5}$ W/m². What is the decibel level?
$$L = 10 \log_{10}\!\left(\frac{5 \times 10^{-5}}{10^{-12}}\right) = 10 \log_{10}(5 \times 10^{7})$$
$$= 10 (\log 5 + 7) = 10(0.699 + 7) = 10(7.699) = 77.0 \text{ dB}$$
About as loud as a busy street. Logarithms and scientific notation give you the answer.
Adding Sound Sources
Two identical speakers don't give you double the decibels — they give you:
$$L_{\text{total}} = L_1 + 10 \log_{10}(2) \approx L_1 + 3 \text{ dB}$$
This is why adding a second speaker to a stereo system doesn't feel twice as loud.
The decibel scale is a logarithm — the same function from your math class. It compresses trillion-fold ranges into a human-readable scale. Logarithms appear whenever nature works across many orders of magnitude.