Question

The intensity, or loudness, of a sound can be measured in decibels (dB), according to the equation I (d B) = 10 log left-bracket StartFraction I Over I Subscript 0 Baseline EndFraction Right-bracket, where I is the intensity of a given sound and I0 is the threshold of hearing intensity. What is the intensity, in decibels, [I(dB)], when I = 10 Superscript 32 Baseline (I Subscript 0)?

15
32
320
737

Answer 1

The intensity of a sound, denoted as I(dB), can be calculated using the formula: \(I(dB) = 10 \log_{10}\left(\frac{I}{I_0}\right)\), where I is the sound intensity, and \(I_0\) is the threshold of hearing intensity.

If \(I = 10^{32}\) and \(I_0 = 10^{32}\), we can substitute these values into the formula:

\[I(dB) = 10 \log_{10}\left(\frac{10^{32}}{10^{32}}\right)\]

Simplifying this expression:

\[I(dB) = 10 \log_{10}(1)\]

Since the logarithm of 1 to any base is always 0, the intensity in decibels (\(I(dB)\)) is 0.

Therefore, the intensity, in decibels, when \(I = 10^{32}\) (with \(I_0 = 10^{32}\)) is 0 dB.