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The loudness of sound (L, measured in decibels) perceived by the human ear depends on intensity levels (I) according to , where
is the standard threshold of audibility. If
then using the change-of-base formula, we get
. At what rate is the loudness changing with respect to x when the intensity is 1,000 times the standard threshold of audibility (that is, when x = 1,000) ?
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