The given formula for decibels (dB) is:

$D(x) = 10 \cdot \log(10^{16} \cdot x),$

where $x$ is the intensity of sound in watts per square meter. We need to find the decibel level when $x = 10^{-4}$.

Step 1: Substitute $x = 10^{-4}$ into the formula

$D(x) = 10 \cdot \log(10^{16} \cdot 10^{-4})$

Step 2: Simplify the expression inside the logarithm

$10^{16} \cdot 10^{-4} = 10^{16 - 4} = 10^{12}$

Thus, $D(x) = 10 \cdot \log(10^{12})$

Step 3: Use the logarithmic property $\log(10^a) = a$

$\log(10^{12}) = 12$

Step 4: Calculate the decibel level

$D(x) = 10 \cdot 12 = 120 \, \text{dB}.$

Final Answer:

The sound level is 120 decibels when the intensity $x = 10^{-4}$.

Would you like a detailed explanation of logarithmic properties or further clarification?

Related Questions:

1. How does increasing the intensity $x$ affect the decibel level?
2. What is the decibel level for $x = 10^{-6}$?
3. Can you derive the general relationship between $x$ and $D(x)$?
4. What happens to the decibel level when $x = 1$?
5. How is the decibel formula used in real-life sound intensity measurements?

Tip:

Remember that the logarithmic scale is nonlinear; a small increase in intensity can result in a significant change in decibels!