What is a decibel?
The decibel (dB) is the fundamental unit for measuring sound intensity, yet it's often misunderstood. Unlike linear measurements, decibels use a logarithmic scale that perfectly matches how humans perceive changes in sound volume. This guide breaks down the complex mathematics into clear, practical concepts anyone can understand.
To grasp decibels, we first need to understand two key mathematical concepts:
Exponents show repeated multiplication: \[ 100 = 10^2 = 10\times10 \] The exponent (2) indicates how many times to multiply the base number (10) by itself. This becomes powerful when dealing with sound intensities that span trillion-fold differences.
Logarithms answer the question: "What power must we raise the base to get this number?" \[ \log_{10} 1000 = 3 \] This means 10 must be multiplied by itself 3 times to reach 1000. Our ears naturally perceive sound in this logarithmic fashion, making decibels the perfect measurement scale.
The decibel is one-tenth of a bel, named after Alexander Graham Bell. It's calculated using: \[ I(dB) = 10\log_{10}\left[\frac{I}{I_0}\right] \] Where:
This relative measurement means:
This table shows how the logarithmic decibel scale translates to real-world sounds:
| Sound Source | Intensity Multiplier | Decibel Level |
|---|---|---|
| Human Hearing Threshold | 1× | 0 dB |
| Quiet Library | 100× | 20 dB |
| Normal Conversation | 100,000× | 50 dB |
| City Traffic | 10,000,000× | 70 dB |
| Rock Concert | 1,000,000,000× | 90 dB |
| Jet Takeoff (100m) | 100,000,000,000,000× | 140 dB |
Note: Prolonged exposure above 85 dB can cause hearing damage, while sounds above 120 dB may cause immediate harm.
The decibel's logarithmic nature perfectly matches human hearing because: