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The intensity, or loudness, of a sound can be measured in decibels (dB) according to the equation:

[tex]\[ I(dB) = 10 \log \left(\frac{I}{I_0}\right) \][/tex]

where [tex]\( I \)[/tex] is the intensity of a given sound and [tex]\( I_0 \)[/tex] is the threshold of hearing intensity.

What is the intensity, in decibels, [tex]\( I(dB) \)[/tex], when [tex]\( I = 10^8 \left(I_0\right) \)[/tex]?

A. 8
B. 9
C. 19
D. 80

Sagot :

GinnyAnsweridn
To determine the intensity of a sound in decibels (dB) given the equation:

[tex]\[ I(dB) = 10 \log \left[ \frac{I}{I_0} \right] \][/tex]

where [tex]\( I \)[/tex] is the intensity of the given sound and [tex]\( I_0 \)[/tex] is the threshold of hearing intensity, follow these steps:

1. Identify the given values:
- [tex]\( I = 10^8 I_0 \)[/tex]
- [tex]\( I_0 \)[/tex] is the threshold of hearing intensity, meaning that [tex]\( I_0 \)[/tex] is a reference value.

2. Substitute the given values into the equation:

[tex]\[ I(dB) = 10 \log \left[ \frac{10^8 I_0}{I_0} \right] \][/tex]

3. Simplify the fraction inside the logarithm:

[tex]\[ \frac{10^8 I_0}{I_0} = 10^8 \][/tex]

4. Rewrite the equation with the simplified fraction:

[tex]\[ I(dB) = 10 \log (10^8) \][/tex]

5. Use the logarithm property [tex]\(\log (10^8) = 8\)[/tex], because the logarithm of a number in base 10, [tex]\( \log_{10} (10^8) \)[/tex], is just the exponent 8:

[tex]\[ I(dB) = 10 \times 8 \][/tex]

6. Perform the multiplication:

[tex]\[ I(dB) = 80 \][/tex]

Hence, the intensity in decibels [tex]\( I(dB) \)[/tex] is [tex]\( 80 \)[/tex].
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