https://file.aiquickdraw.com/mathbot/file/172905170730120q3fd34.jpg

The problem shown in the image is to solve for \( n \) in the following equation:

\[
200(1.04)^n = 11,600
\]

### Step-by-step solution:

1. **Isolate the exponential term**:
   \[
   (1.04)^n = \frac{11,600}{200}
   \]
   Simplifying the right-hand side:
   \[
   (1.04)^n = 58
   \]

2. **Take the natural logarithm (log) of both sides** to solve for \( n \):
   \[
   \ln\left((1.04)^n\right) = \ln(58)
   \]
   Using the logarithmic identity \( \ln(a^b) = b \ln(a) \), we can rewrite the left-hand side:
   \[
   n \ln(1.04) = \ln(58)
   \]

3. **Solve for \( n \)**:
   \[
   n = \frac{\ln(58)}{\ln(1.04)}
   \]

4. **Calculate the result**:
   - \( \ln(58) \approx 4.060443 \)
   - \( \ln(1.04) \approx 0.039221 \)

   Now divide:
   \[
   n \approx \frac{4.060443}{0.039221} \approx 103.52
   \]

So, the value of \( n \) is approximately **103.52**.

Would you like more details or have any questions?

### Relative questions:
1. How do you use logarithmic properties to simplify exponential equations?
2. What is the significance of taking the natural logarithm in exponential equations?
3. How would this process change if the base was different, e.g., \(1.03\) instead of \(1.04\)?
4. Could you solve this equation using logarithms with base 10? How?
5. How would you apply this process to solving compound interest problems?

### Tip:
Always check that your logarithm base matches the base of the exponential you're working with to avoid confusion.

Learn how to solve the exponential equation 200(1.04)^n = 11,600 with step-by-step logarithmic operations. This problem demonstrates key concepts in logarithms and exponential equations, suitable for high school or college-level students.

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