The given formula for loudness level in decibels ($D$) is:

$D = 10 \log_{10} \left( \frac{I}{10^{-12}} \right)$

Where $I$ is the intensity of the sound in watts per square meter. In the problem, $I = 10^{-7} \, \text{W/m}^2$.

Step-by-step Calculation:

1. Substitute $I = 10^{-7}$ into the formula: $D = 10 \log_{10} \left( \frac{10^{-7}}{10^{-12}} \right)$

2. Simplify the fraction: $\frac{10^{-7}}{10^{-12}} = 10^{-7 - (-12)} = 10^{5}$

3. The equation becomes: $D = 10 \log_{10} \left( 10^{5} \right)$

4. Simplify using the logarithmic rule $\log_{10}(10^x) = x$: $D = 10 \cdot 5 = 50$

Final Answer:

The decibel level is: $\boxed{50 \, \text{dB}}$

Would you like further explanation or have questions on logarithmic properties?

Here are five follow-up questions you might explore:

1. What is the relationship between sound intensity and decibel levels?
2. How does doubling the intensity affect the decibel level?
3. Why is $10^{-12}$ used as the reference intensity in this formula?
4. How do logarithmic scales help in measuring sound?
5. Can decibel levels be negative? If so, under what conditions?

Tip: Always check the units of intensity when working with decibel equations to ensure consistency.