Loudspeakers and decibels

Statement

A loudspeaker emits sounds at $80$ dB. How many identical loudspeakers, played simultaneously, are needed to reach $90$ dB ($= 80 + 10$)? And to reach $100$ dB?

Solution

With $n$ loudspeakers the intensity is $n$ times that of a single one, so the level increases by $\Delta L = 10\log_{10} n \ \text{dB}.$

• For $+10$ dB: $10\log_{10} n = 10 \iff \log_{10} n = 1 \iff n = 10$.
• For $+20$ dB: $10\log_{10} n = 20 \iff \log_{10} n = 2 \iff n = 100$.

You need $\boxed{10}$ loudspeakers to reach $90$ dB and $\boxed{100}$ to reach $100$ dB.

Links

Topics: Logarithmic function
Concepts: Decibel · Logarithm · Logarithmic scale
Functions: Logarithmic function
Methods: Decibel
Skills: Calculating · Modelling
Exercise type: Modelling problem