Most acoustical data uses „dB“ scales. Therefore, it is unavoidable to do the logarithmic dance, in order to interpret measurements. Let us take a short look at it.

A perfect human hearing resolves acoustical powers around a threshold of 10^-12 W (0,000000000001 W). The pain and damage limited ceiling is arrived at a few watts of acoustical power. To facilitate such a dynamic range, our perception of loudness works not in a linear, but logarithmic way. Contrasts on low levels are expanded, on high levels compressed.

Logarithmic perception

If acoustical power is shown on a linear x-axis, our corresponding loudness perception looks similar to above plot of the logarithm (base 10).

Besides the expander / compressor function of the logarithm, it cuts the x-axis at x=1:
log(1) = 0

Let us now further adapt the sound pressure level (SPL) scale to our hearing: We reference that logarithm of acoustical power to our hearing threshold. And increase the resolution to better correlate with our hearing. Instead of using „Bel“ the „deci“ (1/10) Bel is introduced by multiplication with a factor of 10:

dB(SPL) = 10 * log(acoustical power / 10^-12)

Now our scale starts with 0 dB(SPL) at the threshold of hearing, up to around 130 dB (SPL) where sounds hurt badly. Changes of about 1 dB are audible increments, while a 10 dB increase equals double perceived loudness. (But corresponds to 10 times of acoustical power.)

Don’t worry too much about the logarithm, it’s simply a way to bend the curve of measured power into the shape of our perception.

How about a virtual test the of the formula?
Imagine to crank up the volume of your music system to double the power output: An SPL meter will show a change of +3 dB. The ratio of power after / before is 2. Now use your calculator: 10 * log(2) = + 3.01 dB

Unsensitive loudspeakers

Some examples of everyday auditory events and their acoustical levels:
Quiet residence 40 dB, office and low level conversation 60 dB, loud traffic noise 90dB.
Live concert levels average around 80 dB for quiet passages, and peak at about 120 dB at forte-fortissimo levels of large orchestra or public address systems.
Even a loud SPL of 120 dB represents only 1 acoustical Watt at our ears!

While the SPL of auditory events are of relatively low power, loudspeakers need a lot more electrical energy to produce those fractions of acoustical watt.
The average modern hifi loudspeaker produces around 85 dB at 1 meter distance from 1 Watt of electrical power input. To increase SPL up to live concert maxima would need 35 dB more power: 3162.3 Watt! Most domestic speaker systems will be destroyed by such input power levels, IF the amplifier can produce them.

But, just like real-size large orchestras don’t fit into our living rooms, we are usually satisfied with „life-like“ reproduction levels of 80-100 dB at home. For which our average hifi loudspeaker needs 0.316 – 31.6 W at 1 m distance. A pair of these loudspeakers at a listening distance of 3 m within a somewhat reflective environment produces 100 dB from an input of about 71 watt per stereo channel. Which is just what many modern amplifiers can provide.

A word of caution

Keep in mind that some music may contain short term peaks of > +10 dB.
And upping the party music level from 100 dB(SPL) to only double perceived loudness equals 10 dB more energy: 10 times more electrical power! Very few domestic systems can do that.
Beware: short exposition to SPL’s of 120 and above cause permanent damage to the human ear!