Example 9 Comparing Sound Intensities

Audio system 1 produces an intensity level of b₁=90.0 dB, and system 2 produces an intensity level of b₂=93.0 dB. The corresponding intensities (in W/m²) are I₁ and I₂. Determine the ratio I₂/I₁.

Reasoning Intensity levels are related to intensities by logarithms (see Equation 16.10), and it is a property of logarithms (see Appendix D) that log A–log B=log (A/B). Subtracting the two intensity levels and using this property, we find that

Solution Using the result just obtained, we find

Doubling the intensity changes the loudness by only a small amount (3 dB) and does not double it, so there is no simple proportionality between intensity and loudness.