Why Are Decibels Logarithmic? The Definition, Exactly

A level in decibels is always a comparison. For a power-like quantity P against a reference P₀, the level is 10 · log₁₀(P / P₀). For a field quantity such as sound pressure p against a reference p₀, the level is 20 · log₁₀(p / p₀).

The factor of twenty is not a separate convention. Acoustic power is proportional to pressure squared, and taking the logarithm turns that square into a factor of two, so 10 · log₁₀(p² / p₀²) becomes 20 · log₁₀(p / p₀). Both expressions describe the same underlying level.

Sound pressure level uses a reference of 20 micropascals, chosen historically as roughly the quietest 1 kHz tone a healthy young ear can detect. Because that reference is fixed by agreement, dB SPL behaves like an absolute figure even though the decibel itself is a ratio.

The pressures involved in hearing span an enormous range — from the threshold of audibility to sounds that cause pain is a factor of millions in pressure. A linear scale would need many digits to express ordinary situations and would make comparison awkward.

A logarithmic scale compresses that range into a couple of hundred convenient units, and it has a second advantage: multiplicative changes become additive. Attenuation through a wall, gain in an amplifier, and loss over distance all simply add and subtract.

Every rule of thumb in acoustics follows from the definition. None of them needs to be memorised separately.

Two independent sources of the same level do not produce twice the level. They produce twice the energy, which is an increase of about 3 dB. So two machines at 80 dB each give roughly 83 dB, not 160.

This is also why the loudest source usually dominates. Add a 70 dB source to an 80 dB source and the result is barely above 80 dB, because the quieter one contributes a small fraction of the total energy.

Psychoacoustic research from the mid-twentieth century onward established that perceived loudness grows more slowly than physical intensity. The classic result is a compressive power relationship, with the sone scale anchored so that one sone equals a 40-phon tone at 1 kHz, and a common summary is that around 10 dB corresponds to a subjective doubling of loudness for a mid-frequency tone.

That summary is a description of typical listeners, not a law. It varies with frequency, level, duration and individual, which is why frequency weightings such as A-weighting exist in the first place: to approximate the ear's uneven sensitivity when a single number is needed.