To solve this problem, we'll use the formula for calculating the intensity of sound in decibels:
[tex]\[ I(dB) = 10 \log_{10}\left(\frac{I}{I_0}\right) \][/tex]
Given:
- [tex]\( I = 10^2 \cdot I_0 \)[/tex]
- [tex]\( I_0 \)[/tex] is the threshold of hearing intensity.
Step-by-step, we can substitute the given values into the formula:
1. Substitute [tex]\( I \)[/tex] with [tex]\( 10^2 \cdot I_0 \)[/tex]:
[tex]\[ I(dB) = 10 \log_{10}\left(\frac{10^2 \cdot I_0}{I_0}\right) \][/tex]
2. Simplify the fraction inside the logarithm:
[tex]\[ I(dB) = 10 \log_{10}(10^2) \][/tex]
3. Recognize that the [tex]\( I_0 \)[/tex] terms cancel each other out:
[tex]\[ I(dB) = 10 \log_{10}(10^2) \][/tex]
4. Apply the properties of logarithms. The logarithm of a power can be written as the exponent times the logarithm of the base:
[tex]\[ \log_{10}(10^2) = 2 \][/tex]
5. Multiply the result by 10:
[tex]\[ I(dB) = 10 \cdot 2 \][/tex]
6. Simplify the multiplication:
[tex]\[ I(dB) = 20 \][/tex]
Therefore, the intensity in decibels is [tex]\( 20 \, dB \)[/tex].
