To solve the equation \( 10,000 = 1500 e^{0.11x} \), follow these steps:

1. **Isolate the exponential term**:
   \[
   e^{0.11x} = \frac{10,000}{1500}
   \]

   Simplifying the right side:
   \[
   e^{0.11x} = \frac{10000}{1500} \approx 6.6667
   \]

2. **Take the natural logarithm of both sides**:
   \[
   \ln(e^{0.11x}) = \ln(6.6667)
   \]

   Using the property of logarithms, this simplifies to:
   \[
   0.11x = \ln(6.6667)
   \]

3. **Solve for \( x \)**:
   \[
   x = \frac{\ln(6.6667)}{0.11}
   \]

   Calculating the natural logarithm:
   \[
   x \approx \frac{1.8971}{0.11} \approx 17.248
   \]

Thus, the solution is approximately \( x \approx 17.25 \).

Would you like more details or have any questions? Here are some related questions you might consider:

1. How do you solve equations involving exponential functions?
2. What are the properties of logarithms that are useful in solving equations?
3. Can you explain the concept of continuous growth represented by exponential functions?
4. How would the solution change if the base of the exponential was different?
5. What practical applications do exponential equations have in real life?

**Tip**: Always isolate the exponential term first when dealing with exponential equations to simplify the solving process.

Discover how to solve the exponential equation 10,000 = 1500e^(0.11x) through a detailed, step-by-step approach. This solution involves isolating exponential terms, applying natural logarithms, and solving for x. Ideal for students in Grades 10-12.

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