In Sound: Pitch and Loudness, loudness came from amplitude. Physics measures it as intensity, power per area. Your ears handle a range of intensities of about a trillion to one, so we squeeze that range with What a Logarithm Is.
Key idea
A source of power $P$ spreading in all directions gives $I = \dfrac{P}{4\pi r^{2}}$ at distance $r$. The sound level is $$\beta = 10\log_{10}\frac{I}{I_0}\ \text{dB}, \qquad I_0 = 10^{-12}\ \text{W/m}^2.$$ Every $+10\ \text{dB}$ means $10$ times the intensity; $+3\ \text{dB}$ is about double.
Why it works
The power spreads over a sphere of area $4\pi r^{2}$, so twice as far means four times the area and a quarter of the intensity. For decibels: $\log_{10}$ turns multiplying into adding. Multiply $I$ by 10 and $\log_{10}(I/I_0)$ goes up by 1, so $\beta$ goes up by 10. Multiply by 2 and $\beta$ goes up by $10\log_{10}2 = 3.0\ \text{dB}$. $I_0$ is roughly the faintest sound a young ear can hear, so $0\ \text{dB}$ is the threshold of hearing, not silence.
Common mistake
Thinking that doubling the intensity doubles the decibel level. Two identical $60\ \text{dB}$ speakers give $63\ \text{dB}$, not $120\ \text{dB}$. Check: $120\ \text{dB}$ is a million times $60\ \text{dB}$ in intensity, far more than two speakers can add.
Worked example: From intensity to decibels
A sound has intensity $1.0 \times 10^{-5}\ \text{W/m}^2$. Find its level.
- $\dfrac{I}{I_0} = \dfrac{10^{-5}}{10^{-12}} = 10^{7}$.
- $\beta = 10\log_{10}(10^{7}) = 10(7) = 70\ \text{dB}$.
Answer: $70\ \text{dB}$
Worked example: Moving away
You hear a small speaker at $80\ \text{dB}$ from $2.0\ \text{m}$ away. What level do you hear at $20\ \text{m}$?
- Ten times as far gives $\frac{1}{10^{2}} = \frac{1}{100}$ of the intensity.
- Dividing by 100 is two factors of 10: $-20\ \text{dB}$.
- $80 - 20 = 60\ \text{dB}$.
Answer: $60\ \text{dB}$
Worked example: Power of a source
At $5.0\ \text{m}$ from a source the intensity is $2.0 \times 10^{-4}\ \text{W/m}^2$. What power does it put out as sound?
- $$P = I \cdot 4\pi r^{2} = (2.0 \times 10^{-4})(4\pi)(5.0)^{2}.$$
- $P = 0.063\ \text{W}$. Sound carries surprisingly little power.
Answer: $0.063\ \text{W}$