To solve the equation \( e^{-0.37x} = 0.88 \) for \( x \), follow these steps:

1. **Take the natural logarithm (ln) of both sides**:
   \[
   \ln\left(e^{-0.37x}\right) = \ln(0.88)
   \]

2. **Simplify the left side**:
   Since \( \ln(e^y) = y \), we have:
   \[
   -0.37x = \ln(0.88)
   \]

3. **Solve for \( x \)**:
   \[
   x = \frac{\ln(0.88)}{-0.37}
   \]

4. **Calculate the value**:
   \[
   x \approx \frac{-0.1278}{-0.37} \approx 0.3457
   \]

So, the solution is \( x \approx 0.3457 \).

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:
1. How would the solution change if the exponent were positive?
2. What is the effect of different coefficients in the exponent on the solution?
3. How do you solve similar equations involving different bases, like \( 2^{-0.5x} = 0.6 \)?
4. What is the general process for solving exponential equations with different constants?
5. How can you apply this to real-world exponential decay problems?

**Tip:** Remember that taking the natural logarithm is a common method for solving equations involving exponentials.

Learn how to solve the exponential equation e^-0.37x = 0.88 with a step-by-step solution. This problem involves exponential functions and natural logarithms. Suitable for students in Grades 10-12.

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