How Long Can You Listen at That Volume?

Sound level in decibels is a logarithm of intensity: every 10 dB means ten times the intensity. Hearing-safety guidelines allow 8 hours a day at 85 dB and halve the time for every 3 dB louder, so finding a safe volume means solving a logarithmic equation.

How it works

Logarithms and logarithmic equations

In one sentence

The logarithm $\log_b x$ is the exponent $y$ with $b^y = x$ , so $\log_b b^u = u$ and $b^{\log_b x} = x$ . To solve $8/2^{(L-85)/3} = t$ , isolate the exponential, $2^{(L-85)/3} = 8/t$ , then take $\log_2$ : $(L - 85)/3 = \log_2(8/t)$ .

General formula

{\htmlData{sym=0}{L}} = 10\log_{10}\frac{\htmlData{sym=1}{{I}}}{{\htmlData{sym=2}{I_0}}}, \qquad {\htmlData{sym=3}{t}} = \frac{8}{2^{({\htmlData{sym=0}{L}} - 85)/3}}, \qquad {\htmlData{sym=0}{L}} = 85 + 3\log_2\frac{8}{\htmlData{sym=3}{{t}}}

Each 3 dB step doubles the sound energy, so the safe time halves. A logarithm counts how many doublings separate two levels.

What the symbols mean

$L$
Sound level
Loudness in decibels.
$I$
Intensity
The sound's power per square meter.
$I_0$
Threshold of hearing
The faintest intensity a typical ear can detect.
$t$
Safe listening time
Hours per day before hearing damage becomes likely.

Decibels compress a huge range: $L = 10\log_{10}(I/I_0)$ , where $I_0$ is the faintest sound you can hear. Every +10 dB multiplies the intensity by 10, and every +3 dB roughly doubles it.
The NIOSH guideline allows 8 hours at 85 dB and halves the time for each 3 dB more: $t = 8/2^{(L - 85)/3}$ hours, an exponential function of $L$ .
To find the loudest safe level for a listening time $t$ , isolate the exponential, $2^{(L - 85)/3} = 8/t$ , take $\log_2$ of both sides and solve: $L = 85 + 3\log_2(8/t)$ .
Logarithms undo exponentials: $\log_2 2^u = u$ is what brings the exponent down to where you can solve for it.

Learn the math

Logarithm FunctionsAlgebra · Lesson 40, Exponential and Logarithm Functions Solving Logarithm EquationsAlgebra · Lesson 42, Exponential and Logarithm Functions Applications of Exponentials and LogarithmsAlgebra · Lesson 43, Exponential and Logarithm Functions