You are trying to decide between two new stereo amplifiers. One is rated at 75 W per channel and the other is rated at 120 W per channel. In terms of dB, how much louder will the more powerful amplifier be when both are producing sound at their maximum levels?
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This is Giancoli Answers with Mr. Dychko. The difference in the decibels for these 2 systems is going to be the decibel rating for the 120 watt system which will be 10 times logarithm of its intensity divided by some reference intensity at the threshold of hearing. Minus 10 times logarithm of the intensity of the 75 watt system divided by the same threshold of hearing I naught and we can write this as 10 times log of... Well, is factored out from both terms. So, we have 10 times log of I 120 minus log I naught, where I have taken this logarithm of a fraction as logarithm of the top thing minus logarithm of the bottom thing, that's one of the, that's the difference rule for logarithms. And then minus log I 75 minus log I naught, applying the same rule to this logarithm here. And then these two things are the same with opposite signs as they disappear. So, we have 10 times log I 120 minus log I 75. So, the intensity is watts per meter squared, but we're going to be using the same speakers for the two different amplifiers presumably. So, the speakers will have the same area on the speaker cones. And so it's enough to just compare their power outputs. So, I mean, this is really watts per area, but since we have the same area for both of them, we could apply the same sort of difference rule here for logarithms and have log 120 watts minus log of whatever this area is. And then minus log 75 minus log of the area, and you'd have the same sort of pattern here showing up in the log of the areas would cancel away anyhow. So, we have 2.0 decibels would be the difference in decibel ratings for these 2 amplifiers, which is a really small difference.
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