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Natural log (ln) is not allowed to be used in this assignment.

To solve the equation \(3^{n+5} - 1 = 177146\), we'll use properties of exponents and logarithms, but since natural logarithms are not allowed, we will use base-10 logarithms (common logarithms) instead.

### Step-by-Step Solution:

1. **Isolate the exponential term:**
   \[
   3^{n+5} = 177146 + 1 = 177147
   \]

2. **Take the logarithm of both sides (base 10):**
   \[
   \log_{10}(3^{n+5}) = \log_{10}(177147)
   \]

3. **Apply the power rule of logarithms:**
   \[
   (n+5) \log_{10}(3) = \log_{10}(177147)
   \]

4. **Solve for \(n\):**
   \[
   n+5 = \frac{\log_{10}(177147)}{\log_{10}(3)}
   \]
   \[
   n = \frac{\log_{10}(177147)}{\log_{10}(3)} - 5
   \]

### Calculation:

1. **Find \(\log_{10}(177147)\):**
   \[
   \log_{10}(177147) \approx 5.2482
   \]

2. **Find \(\log_{10}(3)\):**
   \[
   \log_{10}(3) \approx 0.4771
   \]

3. **Calculate \(n\):**
   \[
   n = \frac{5.2482}{0.4771} - 5
   \]
   \[
   n \approx \frac{5.2482}{0.4771} - 5
   \]
   \[
   n \approx 11 - 5
   \]
   \[
   n \approx 6
   \]

Thus, the value of \(n\) is 6.

### Summary:
- The equation \(3^{n+5} - 1 = 177146\) was solved using common logarithms.
- The exponent rule and the logarithm power rule were applied.
- The value of \(n\) was found to be \(n = 6\).

Would you like further details or have any questions?

### Additional Questions:

1. How do logarithm properties help in solving exponential equations?
2. What are the common mistakes when dealing with logarithmic equations?
3. Can you solve the equation using another method without logarithms?
4. What is the importance of isolating the exponential term before applying logarithms?
5. How would you explain the power rule of logarithms?
6. What are some applications of exponential equations in real life?
7. Why is it important to follow the steps methodically in logarithmic equations?
8. What is the relationship between logarithms of different bases?

### Tip:
When working with logarithms, always ensure that the base you are using is appropriate for the context, and double-check your calculations to avoid errors.

Learn how to solve the exponential equation 3^(n+5) - 1 = 177146 using logarithms. This problem involves properties of exponents and logarithms, focusing on the common logarithm (base-10 logarithm) due to restrictions on natural logarithms. Suitable for high school students studying algebra and exponential functions.

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