To determine the increase in intensity level when the intensity of sound increases by a factor of [tex]\(10^5\)[/tex], we follow these steps:
1. Understand the Relationship: The relationship between the intensity increase factor and the increase in intensity level (in decibels, dB) is defined by the formula:
[tex]\[
\text{Increase in dB} = 10 \times \log_{10}(\text{intensity increase factor})
\][/tex]
2. Given Intensity Increase Factor: The intensity increase factor is given as [tex]\(10^5\)[/tex].
3. Substitute and Calculate:
- Substitute the given intensity increase factor into the formula:
[tex]\[
\text{Increase in dB} = 10 \times \log_{10}(10^5)
\][/tex]
- Evaluate [tex]\(\log_{10}(10^5)\)[/tex]. Since [tex]\(\log_{10}(10^5)\)[/tex] simplifies to 5 (because the logarithm base 10 of [tex]\(10^5\)[/tex] is the exponent 5):
[tex]\[
\log_{10}(10^5) = 5
\][/tex]
- Now, multiply this result by 10:
[tex]\[
\text{Increase in dB} = 10 \times 5 = 50 \text{ dB}
\][/tex]
4. Conclusion: Therefore, the increase in intensity level when the intensity increases by a factor of [tex]\(10^5\)[/tex] is [tex]\(50 \text{ dB}\)[/tex].
So, the correct answer is:
[tex]\[ \boxed{50 \text{ dB}} \][/tex]
