How a Decibel Adds Up: Logarithms, +3 dB, and Safe Noise

The first time someone tells you that two 60 dB speakers playing together measure 63 dB and not 120, it sounds like a trick. It is not. It is the whole point of the decibel, and once the logarithm clicks into place, a pile of noise problems that used to feel like guesswork turn into one short calculation. This post walks through why the scale behaves the way it does, what +3 dB and +10 dB really cost in power, how to combine sources correctly, and where the safe-exposure lines actually sit.

The decibel is a ratio, not a count

A decibel is not a unit of sound the way a meter is a unit of distance. It is a way of expressing a ratio between two quantities on a logarithmic scale. For power, the definition is:

L = 10 · log10(P2 / P1)

That single formula carries every surprising property the decibel has. Because it is logarithmic, equal multiplications of power show up as equal additions of decibels. Multiply the power by 10 and you add exactly 10 dB, every time, because 10 · log10(10) = 10. Multiply it by 100 and you add 20 dB. Multiply it by 1000 and you add 30 dB. The scale compresses an enormous range — from the rustle of leaves to a jet engine spans about a trillion to one in power — into a tidy span of roughly 0 to 140.

This is also why you cannot read decibels with arithmetic intuition. The gap from 30 dB to 40 dB is not "ten more of something." It is ten times the power. The gap from 40 to 50 is another ten times. By the time you climb from 30 dB to 90 dB you have multiplied the power by a million, not added sixty.

Why +10 dB is ten times and +3 dB is double

Two numbers are worth memorizing, because almost everything else follows from them.

+10 dB is 10× the power. This falls straight out of the definition, since 10 · log10(10) = 10 dB exactly. Add 10 dB and you have ten times the acoustic power. Drop 10 dB and you have a tenth.

+3 dB is roughly double the power. A power ratio of 2 gives 10 · log10(2) = 3.01 dB. So every time you double the power, you climb about 3 dB; every time you halve it, you fall about 3 dB. That is why audio engineers treat 3 dB as a natural step size: it is the doubling point.

Loudness as the ear judges it is a softer, more subjective thing, but the rough rule of thumb most people use is that +10 dB sounds about twice as loud, while +3 dB is a clear, noticeable jump rather than a dramatic one. The physics (power) and the perception (loudness) run on different curves, and conflating them is where a lot of confusion starts.

One more fork worth flagging: when your ratio is an amplitude — voltage, current, or sound pressure — you use 20 · log10 instead of 10 · log10, because power is proportional to amplitude squared and the square pulls out as a factor of two. A voltage ratio of 2 is 6.02 dB, while a power ratio of 2 is only 3.01 dB. Pick the wrong formula and your answer is off by exactly double. If you want to feel how that 20·log path behaves against the 10·log one, the decibel calculator lets you switch modes and watch the same ratio land on two different numbers.

Adding sound sources: the worked example

Here is the part that breaks intuition, and the part the math fixes cleanly. To combine independent (uncorrelated) sources, you do not add their decibel figures. You convert each level back to a power value, sum the powers, and convert the total back to decibels:

L_total = 10 · log10( 10^(L1/10) + 10^(L2/10) + ... )

Take two equal sources at 60 dB. Each one represents a power value of 10^(60/10) = 10^6. Two of them sum to 2 × 10^6. Convert back:

L = 10 · log10(2 × 10^6) = 10 · log10(2) + 10 · log10(10^6) = 3.01 + 60 = 63.01 dB

So two equal 60 dB sources combine to about 63 dB, not 120. The reason is the same +3 dB rule from earlier: two equal sources double the total power, and doubling power is +3 dB. Add a third equal source and the total is three times the original power, which is +4.77 dB, landing at about 64.8 dB. Ten equal sources are ten times the power, exactly +10 dB, at 70. The additions get smaller and smaller because each new copy is a smaller fraction of a growing total.

Unequal sources behave even more sharply. A loud source dominates and a quiet one barely registers. An 85 dB machine next to a 60 dB fan combines to about 85.01 dB — the fan's contribution is so small it is lost in rounding. This is the single most useful consequence of the logarithm for anyone doing noise control: chasing the quiet contributors is wasted effort, and the loudest source sets the room.

Where I learned to trust the math

I spent an afternoon in a print shop trying to talk a manager out of buying a quieter second cutter. He was sure that adding any machine would stack noise toward the danger zone, the way two of anything "adds up." We measured the running cutter at 84 dB and the proposed replacement spec'd at 78 dB. Combined, that is about 85 dB — a single decibel over the one machine, because the quieter unit contributes so little power. The expensive low-noise model would have changed the reading by a rounding error. What actually moved the number was an enclosure on the loud cutter, which knocked it down 8 dB and dragged the combined level with it. Watching the manager's face when the second machine "did nothing" to the meter is the moment the logarithm stopped being abstract for me. I have reached for that 10 · log10 sum on every noise question since.

Safe noise exposure and the dose trade

Decibels matter most when they cross health thresholds. The widely cited workplace guidance treats 85 dB as the action level for an 8-hour shift — the point where hearing protection and monitoring come into play. Above that, the exposure budget shrinks fast, and it shrinks on the same logarithmic logic as everything else.

The common exchange rate is that every 3 dB doubles the sound energy, so it halves the time you can safely absorb it. If 85 dB is safe for 8 hours, then 88 dB is safe for 4, 91 dB for 2, 94 dB for 1, and so on. (Some standards use a 5 dB exchange rate instead, which is more lenient; the 3 dB rule tracks the actual energy and is the stricter, more physically honest choice.) A rock concert at 100 dB or a chainsaw at 110 dB blows through a full day's dose in minutes. This is why a brief exposure to a very loud sound and a long exposure to a moderately loud one can carry the same risk — the calculation is a product of level and time, and the level term is logarithmic.

If you want to sanity-check one of these conversions by hand — turning a 3 dB step into a power-of-two, or a 10 dB step into a power-of-ten — a general-purpose scientific calculator with a log button will reproduce every figure in this post. The arithmetic is short; the only trap is remembering to take powers before you sum and logs after.

The short version

Decibels are ratios on a logarithmic scale, so they multiply where ordinary numbers add. +10 dB is ten times the power, +3 dB is double, and doubling the number of equal sources adds about 3 dB. To combine sources, sum the powers and convert back — which is why two 60 dB sources make 63 dB, and why a quiet machine next to a loud one barely moves the needle. Keep those rules in mind and noise stops being mysterious. It becomes one logarithm you can do on the back of an envelope, and one threshold — 85 dB for 8 hours — that tells you when to stop guessing and start protecting your ears.

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Made by Toolora · Updated 2026-06-13