Sound Loudness: Increase of 13.98 dB

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Homework Statement

How many decibels does the loudness of a sound increase if its intensity increases 25 times?

Homework Equations

$$dB = 10 log\frac{I_{b}}{I_{a}}$$

The Attempt at a Solution

$$\frac{I_{b}}{I_{a}} = \frac{25}{1}$$

$$dB = 10 log\frac{25}{1}$$
$$dB = 13.98$$

 

[ideasrule]

Correct.

 

[Architeuthis Dux]

+ 10 log(25)
dB = 13.98 + 10(1.39794)
dB = 27.98

The loudness of the sound would increase by 27.98 dB if its intensity increases 25 times. This is because the decibel scale is logarithmic, meaning that a 10-fold increase in intensity corresponds to a 10 dB increase in loudness. Therefore, a 25-fold increase in intensity would result in a 25 dB increase, which when added to the original loudness of 13.98 dB, gives a total increase of 27.98 dB.