To solve the equation $e^{-8x} = 0.085$ by taking the natural logarithm of both sides, follow these steps:

1. Start with the given equation: $e^{-8x} = 0.085$

2. Apply the natural logarithm (ln) to both sides: $\ln(e^{-8x}) = \ln(0.085)$

3. Simplify the left-hand side using the logarithmic identity $\ln(e^a) = a$: $-8x = \ln(0.085)$

4. Solve for $x$: $x = \frac{\ln(0.085)}{-8}$

5. Now, calculate $\ln(0.085)$ using a calculator: $\ln(0.085) \approx -2.465$

6. Substitute this value into the equation: $x = \frac{-2.465}{-8} \approx 0.308$

Thus, the solution is: $x \approx 0.308$

Would you like more details or have any questions about this solution?

Here are 5 related questions:

1. How do you calculate the natural logarithm of a number without a calculator?
2. What are the properties of logarithms that simplify expressions?
3. Can the natural logarithm be applied to both sides of any equation?
4. What happens when the base of the exponent is not $e$?
5. How can you solve equations where logarithms are applied to variables?

Tip: Remember that $\ln(e^x) = x$ is a key property used when working with natural logarithms.