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].
