Decibel Formula
Reference for dB = 10 log10(I/I0) and dB = 20 log10(P/P0).
Covers intensity, pressure, and power ratios with a table from 0 dB (hearing) to 140 dB (jet).
Need to calculate, not just reference? Use the interactive version.
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The Formula
dB = 10 × log₁₀(I / I₀)
The decibel scale measures sound intensity on a logarithmic scale. Every 10 dB increase represents a tenfold increase in sound intensity.
Variables
| Symbol | Meaning |
|---|---|
| dB | Sound level in decibels |
| I | Sound intensity being measured (W/m²) |
| I₀ | Reference intensity — threshold of hearing (10⁻¹² W/m²) |
| log₁₀ | Common logarithm (base 10) |
Example 1
A sound has intensity 10⁻⁶ W/m². What is its decibel level?
dB = 10 × log₁₀(10⁻⁶ / 10⁻¹²)
dB = 10 × log₁₀(10⁶) = 10 × 6
dB = 60 dB (normal conversation level)
Example 2
How many decibels is a sound 1,000 times more intense than another at 50 dB?
Additional dB = 10 × log₁₀(1,000) = 10 × 3 = 30
New level = 50 + 30 = 80 dB
When to Use It
Use the decibel formula when:
- Measuring and comparing sound levels
- Assessing noise exposure for safety standards
- Designing audio systems and soundproofing
- Converting between linear intensity and the logarithmic dB scale
Key Notes
- For sound pressure (Pa), use dB = 20 × log₁₀(P/P₀) with P₀ = 20 μPa — the factor is 20 (not 10) because intensity is proportional to pressure squared
- Doubling the intensity adds only ~3 dB; to double perceived loudness requires about 10 dB — the scale compresses a trillion-to-one physical range into a 0–130 dB human range
- Decibels are always relative to a reference value — dB SPL, dBm, and dBFS all use different references and cannot be directly compared without knowing which scale is meant
Key Notes
- Sound level: L = 10 log(I/I₀) dB: I₀ = 10⁻¹² W/m² is the threshold of human hearing. For sound pressure: L = 20 log(P/P₀) where P₀ = 20 µPa. The factor 20 (vs 10) appears because intensity is proportional to pressure squared (I ∝ P²).
- Logarithmic scale intuition: +10 dB = 10× the intensity (sounds about twice as loud perceptually). +20 dB = 100× intensity; +30 dB = 1,000× intensity. Two identical sources together add only +3 dB (double the intensity = 10 log 2 ≈ 3 dB), not +6 dB.
- Key reference levels: Threshold of hearing: 0 dB; library: ~40 dB; normal conversation: ~60 dB; busy traffic: ~85 dB (hearing damage threshold for sustained exposure); rock concert: ~110 dB; jet engine at 30 m: ~150 dB.
- Inverse square law: In free field (outdoors), sound intensity drops as 1/r². Doubling distance reduces intensity by 75% — a drop of 6 dB. This is why outdoor noise ordinances use distance-weighted measurements and why concert venue acoustics differ from open fields.
- Applications: Decibel calculations are used in speaker and amplifier specifications, noise pollution regulations, occupational hearing protection (OSHA limits), room acoustic design, audio engineering, and telecommunications (signal-to-noise ratio, attenuation in cables and optical fibers).