Find n to 9 decimal places in the equation: 364.3106 = 425/(1+0.03875)^n

We are given the equation:

\[
364.3106 = \frac{425}{(1 + 0.03875)^n}
\]

We need to solve for \( n \) to 9 decimal places. Let's walk through the steps:

### Step 1: Rearrange the equation to isolate \( (1 + 0.03875)^n \)

\[
(1 + 0.03875)^n = \frac{425}{364.3106}
\]

### Step 2: Simplify the right-hand side

\[
(1 + 0.03875)^n = \frac{425}{364.3106} \approx 1.166686
\]

### Step 3: Apply the logarithm to both sides

Take the natural logarithm (ln) of both sides:

\[
\ln \left( (1 + 0.03875)^n \right) = \ln(1.166686)
\]

\[
n \cdot \ln(1 + 0.03875) = \ln(1.166686)
\]

### Step 4: Solve for \( n \)

Now, divide both sides by \( \ln(1 + 0.03875) \):

\[
n = \frac{\ln(1.166686)}{\ln(1 + 0.03875)}
\]

Let me calculate that.The value of \( n \), to 9 decimal places, is approximately:

\[
n \approx 4.052872222
\]

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

Here are some related questions to consider:
1. How do logarithms simplify solving exponential equations?
2. What other types of growth models use equations similar to this?
3. How does the value of the rate (0.03875) affect the solution?
4. Can the equation be solved without logarithms?
5. How does changing the initial value of 425 affect the result?

**Tip:** When solving exponential equations, logarithms are often the most effective tool to isolate the exponent.

Learn how to find n to 9 decimal places in the exponential equation 364.3106 = 425/(1 + 0.03875)^n. This solution involves key mathematical concepts such as exponential equations and logarithms, demonstrating how to isolate the variable using logarithmic properties. Suitable for students in Grades 10-12.

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