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To solve this problem, we need to determine after how many full years, $ n $, the value of a vintage collector's item will exceed $10,000, given that it costs $4,000 initially and appreciates at a rate of $ \frac{26}{5} \% $ per annum.

### Step-by-step Solution:

1. **Understanding the Problem:**
   - Initial value, $ V_0 = 4000 $ dollars.
   - Annual appreciation rate, $ r = \frac{26}{5} \% = 5.2\% = 0.052 $ (as a decimal).
   - Future value, $ V(t) = V_0(1 + r)^n $.
   - We need to find $ n $ such that $ V(n) > 10000 $.

2. **Set Up the Equation:**
   \[
   4000 \times (1 + 0.052)^n > 10000
   \]

3. **Divide Both Sides by 4000:**
   \[
   (1 + 0.052)^n > \frac{10000}{4000}
   \]
   \[
   (1.052)^n > 2.5
   \]

4. **Apply Logarithms to Solve for $ n $:**
   \[
   \log((1.052)^n) > \log(2.5)
   \]
   \[
   n \cdot \log(1.052) > \log(2.5)
   \]
   \[
   n > \frac{\log(2.5)}{\log(1.052)}
   \]

5. **Calculate the Value of $ n $:**

Let's compute the value.The value of the vintage collector's item will exceed $10,000 after **19 full years**.

### Summary:
- After 19 years of appreciating at a rate of $5.2\%$ per annum, the initial investment of $4,000 will grow to exceed $10,000.

Would you like further details or have any questions about the solution?

### Related Questions:
1. What will be the value of the item after 19 years exactly?
2. How does the appreciation rate affect the time it takes for an investment to double?
3. What is the formula for compound interest, and how does it apply to different compounding intervals?
4. How can logarithms help in solving exponential growth problems in finance?
5. How does changing the initial value affect the time to reach a specific future value?

### Tip:
For financial calculations involving appreciation or depreciation, understanding the compounding effect is crucial, as it can significantly affect the growth of an investment over time.

A vintage collectors item that costs $4000, appreciates at approximately 26/5% per annum. After how many full years, n, will the value of the vintage collectors item be over $10,000?

Learn how to determine after how many years a vintage item that costs $4000 and appreciates at 26/5% per year will exceed $10,000. This solution covers key exponential growth concepts, including the use of logarithms to find the number of years required for the item's value to reach a given amount.

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