Your ear spans a trillion-to-one range of intensities, so loudness is scored on a ladder of tens. The raw quantity is intensity — power spread over area, $\mathrm{W/m^2}$ — and the decibel compares it to the threshold of hearing, $I_0 = 10^{-12}\ \mathrm{W/m^2}$: $\beta = 10\log_{10}\!\left(\dfrac{I}{I_0}\right)$ — read aloud: beta equals ten log I over I-nought. $\beta$ is the Greek letter beta, the sound level in decibels and the thing you are solving for; $I$ is the measured intensity in W/m²; and the subscript nought on $I_0$ marks the fixed threshold everything is compared against.

The whole trick: every power of ten of intensity is worth exactly 10 dB. Count the powers, multiply by ten, done — and doubling the intensity adds only about 3 dB, which is why a second violin barely moves the meter. Order-of-ten thinking IS the calculation here.

Distance fades sound by the inverse-square law: the power spreads over a sphere of area $4\pi r^2$, $r$ being your distance from the source, in metres — so twice as far means a quarter the intensity, about 6 dB down the ladder.